when the asterisk mark appear on the report, what does it mean?
The * shown in the question appears when we should reject the hypothesis of true uniform random input with low probability of false rejection, based on the fraction on the left, where the numerator is the number of "successful" tests, and the denominator is the number of tests, with test's type on the right.
We must turn to the source code to find that the * tells that in this fraction passCount/sampleSize the numerator is out of bounds [proportion_threshold_min, proportion_threshold_max] computed as:
p_hat = 1.0 - ALPHA;
proportion_threshold_max = (p_hat + 3.0 * sqrt((p_hat*ALPHA)/sampleSize)) * sampleSize;
proportion_threshold_min = (p_hat - 3.0 * sqrt((p_hat*ALPHA)/sampleSize)) * sampleSize;
where parameter ALPHA (by default $\alpha=0.01=1\%$) "denotes the significance level that determines the region of acceptance and rejection", and (thus) the expected proportion of tests that fail for true uniform random input. The formulas for these bounds are asymptotically those of the "3-sigma rule" (probability $0.27\%$ that passCount is outside the bounds under a Gaussian assumption of the distribution), with rounding to nearest integer. Notice it takes many tests before that stabilizes! The table below gives the [Min, Max] interval for acceptance (no *), and the residual probability that the * appears for true uniform random input.
Tests 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150 155 160 165 170 175 180 185 190 195 200 205 210 215 220 225 230 235 240 245 250 255 260 265 270 275 280 285 290 295 300 305 310 315 320 325 330 335 340 345 350 355 360 365 370 375 380 385 390 395 400 405 410 415 420 425 430 435 440 445 450 455 460 465 470 475 480 485 490 495 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700 710 720 730 740 750 760 770 780 790 800 810 820 830 840 850 860 870 880 890 900 910 920 930 940 950 960 970 980 990 1000 1010 1020 1030 1040 1050 1060 1070 1080 1090 1100 1110 1120 1130 1140 1150 1160 1170 1180 1190 1200 1210 1220 1230 1240 1250 1260 1270 1280 1290 1300 1310 1320 1330 1340 1350 1360 1370 1380 1390 1400 1410 1420 1430 1440 1450 1460 1470 1480 1490 1500 1510 1520 1530 1540 1550 1560 1570 1580 1590 1600 1610 1620 1630 1640 1650 1660 1670 1680 1690 1700 1710 1720 1730 1740 1750 1760 1770 1780 1790 1800 1810 1820 1830 1840 1850 1860 1870 1880 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000
Min 1 2 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 44 45 46 47 52 57 62 67 72 77 81 86 91 96 101 106 111 116 120 125 130 135 140 145 150 155 160 164 169 174 179 184 189 194 199 204 208 213 218 223 228 233 238 243 248 253 257 262 267 272 277 282 287 292 297 302 307 311 316 321 326 331 336 341 346 351 356 361 365 370 375 380 385 390 395 400 405 410 415 420 424 429 434 439 444 449 454 459 464 469 474 478 483 488 498 508 518 528 537 547 557 567 577 587 597 606 616 626 636 646 656 665 675 685 695 705 715 724 734 744 754 764 774 784 793 803 813 823 833 843 852 862 872 882 892 902 912 921 931 941 951 961 971 981 990 1000 1010 1020 1030 1040 1050 1059 1069 1079 1089 1099 1109 1119 1128 1138 1148 1158 1168 1178 1188 1197 1207 1217 1227 1237 1247 1257 1266 1276 1286 1296 1306 1316 1326 1335 1345 1355 1365 1375 1385 1395 1404 1414 1424 1434 1444 1454 1464 1473 1483 1493 1503 1513 1523 1533 1542 1552 1562 1572 1582 1592 1602 1612 1621 1631 1641 1651 1661 1671 1681 1690 1700 1710 1720 1730 1740 1750 1759 1769 1779 1789 1799 1809 1819 1829 1838 1848 1858 1868 1878 1888 1898 1907 1917 1927 1937 1947 1957 1967
Max 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150 155 160 165 170 175 180 185 190 195 200 205 210 215 220 225 230 235 240 245 250 255 260 265 270 275 280 285 290 295 300 305 310 315 320 325 330 335 340 345 350 355 360 365 370 375 380 385 390 395 400 405 410 415 420 425 430 435 440 445 450 455 460 465 470 475 480 485 490 495 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700 710 720 730 740 750 760 770 780 790 800 810 820 830 840 850 860 870 880 890 900 910 920 930 940 950 960 970 980 989 999 1009 1019 1029 1039 1049 1059 1069 1079 1089 1099 1109 1119 1129 1139 1149 1159 1169 1178 1188 1198 1208 1218 1228 1238 1248 1258 1268 1278 1288 1298 1308 1318 1328 1338 1347 1357 1367 1377 1387 1397 1407 1417 1427 1437 1447 1457 1467 1477 1487 1497 1506 1516 1526 1536 1546 1556 1566 1576 1586 1596 1606 1616 1626 1636 1646 1656 1665 1675 1685 1695 1705 1715 1725 1735 1745 1755 1765 1775 1785 1795 1805 1815 1824 1834 1844 1854 1864 1874 1884 1894 1904 1914 1924 1934 1944 1954 1964 1973 1983 1993
Prob[*] 1.00% 1.99% 0.03% 0.06% 0.10% 0.15% 0.20% 0.27% 0.34% 0.43% 0.52% 0.62% 0.72% 0.84% 0.96% 1.09% 1.23% 1.38% 1.53% 0.10% 0.12% 0.13% 0.15% 0.17% 0.20% 0.22% 0.24% 0.27% 0.30% 0.33% 0.36% 0.40% 0.44% 0.47% 0.52% 0.56% 0.60% 0.65% 0.70% 0.75% 0.80% 0.86% 0.92% 0.98% 1.04% 1.10% 0.13% 0.14% 0.15% 0.16% 0.23% 0.31% 0.42% 0.54% 0.69% 0.87% 0.17% 0.22% 0.28% 0.34% 0.42% 0.51% 0.62% 0.74% 0.17% 0.21% 0.25% 0.30% 0.36% 0.42% 0.49% 0.57% 0.66% 0.17% 0.21% 0.24% 0.28% 0.32% 0.37% 0.43% 0.49% 0.56% 0.16% 0.18% 0.21% 0.24% 0.28% 0.32% 0.36% 0.40% 0.45% 0.51% 0.16% 0.18% 0.20% 0.23% 0.26% 0.29% 0.32% 0.36% 0.40% 0.45% 0.49% 0.16% 0.18% 0.21% 0.23% 0.26% 0.28% 0.31% 0.35% 0.38% 0.42% 0.46% 0.16% 0.18% 0.20% 0.22% 0.24% 0.27% 0.29% 0.32% 0.35% 0.39% 0.42% 0.46% 0.17% 0.19% 0.21% 0.23% 0.25% 0.27% 0.30% 0.32% 0.35% 0.38% 0.41% 0.16% 0.17% 0.19% 0.23% 0.27% 0.31% 0.36% 0.16% 0.19% 0.22% 0.26% 0.30% 0.34% 0.40% 0.18% 0.21% 0.24% 0.28% 0.32% 0.37% 0.17% 0.20% 0.23% 0.26% 0.30% 0.34% 0.16% 0.18% 0.21% 0.24% 0.28% 0.31% 0.35% 0.17% 0.20% 0.22% 0.25% 0.29% 0.32% 0.16% 0.18% 0.20% 0.23% 0.26% 0.29% 0.33% 0.16% 0.19% 0.21% 0.23% 0.26% 0.30% 0.33% 0.17% 0.19% 0.22% 0.24% 0.27% 0.30% 0.33% 0.17% 0.19% 0.21% 0.24% 0.27% 0.29% 0.33% 0.17% 0.19% 0.21% 0.25% 0.27% 0.30% 0.33% 0.18% 0.20% 0.22% 0.24% 0.26% 0.29% 0.32% 0.17% 0.19% 0.21% 0.23% 0.26% 0.28% 0.32% 0.18% 0.20% 0.22% 0.24% 0.26% 0.28% 0.31% 0.17% 0.19% 0.21% 0.23% 0.25% 0.27% 0.29% 0.17% 0.20% 0.21% 0.23% 0.25% 0.27% 0.29% 0.17% 0.18% 0.20% 0.22% 0.23% 0.25% 0.28% 0.30% 0.17% 0.19% 0.22% 0.24% 0.26% 0.28% 0.30% 0.18% 0.19% 0.21% 0.22% 0.24% 0.26% 0.28% 0.17% 0.18% 0.19% 0.21% 0.25% 0.26% 0.28% 0.30% 0.18% 0.20% 0.21% 0.23% 0.24% 0.26% 0.28% 0.17% 0.18% 0.20% 0.21% 0.25% 0.26% 0.28%
In the question's occurence of *, there are $41$ tests, the acceptance interval is $[39,41]$, and the residual probability that the * appears (as it does) is $0.80\%$. However the number of successes is $37$, which is more than one outside the acceptance interval, and the actual probability of such result is
$$1-\sum_{38\le k\le41}{\binom{41}k(1-\alpha)^k \alpha^{41-k}}\quad\approx 0.075\%$$
Even considering that's one out of four RandomExcursionsVariant series of tests, it's a reason for alarm. At least, the series of tests should be repeated with different seed, or much more tests.
There can also be a * on the right of the P-value. Looking at the source tells that's for sampleSize<10. Note 11 in the nice manual correspondingly tells that the P-value "is undefined if less than 10 sequences are processed. As noted Section 4.2 at least 55 sequences must be processed to derive statistically meaningful results for the uniformity of p-values".
Some tests have P-value lower than 0.01. However, the asterisk mark did not appear. Why is that?
That's because the * appears on the basis of the number of failed tests, not based on the P-value shown.
At a different level, that's in some sense because for uniformly random input, the * appears with significantly lower probability ($0.27\%$ asymptotic target, $0.69\%$ for $75$ tests) than the probability ($1\%$) that the P-value is lower than $0.01$ as in the observation made.
Further, I think that the range to be considered should be $[0.005,0.995]$ for the $99\%$ confidence level, and the question's 0.005033 falls within that, if only barely.
Even further, with 6 series of tests and such $99\%$ confidence level on each series, we only have about $94\%$ confidence level that no series fail.
As usual: passing the NIST test, or any statistical test of the generator's output, is not a valid argument towards the suitability of a PRNG or TRNG for cryptographic purposes, contrary to what countless publications pretend or let suppose. As an illustration, NIST's Dual_EC_DRBG passes the NIST test, even though it is experimentally distinguishable from random and (independently) backdoorable by design, thus must be considered insecure, and has been exploited.
