I'm going to give a more detailed version of @poncho's answer. Let me know if you'd like more detail on any point and I can make it still longer. Ideas that you might need to read additional background on are given in italics, but if other sources aren't clear, let me know.
Short Answer
ECDSA is not broken by factoring into primes, rather both problems are solved by variations of the quantum Fourier transform. Although slightly different, these variations are both known as Shor's algorithm.
Notation warning
In the following, I will use a bad habit of mathematicians to use the "+" sign to mean several different things. In all cases, it will be an operation that combines two members of set to make another member of the set. All uses are examples of abelian groups and it is useful to borrow our intuition of how addition works because the operations in our groups will behave similarly. I'll try to distinguish different groups by writing the sets in different ways. Specifically, I will use "+" for:
- Adding real numbers in the usual way. I'll write $\mathbb R$ for the set and lower case Greek letters such as $\alpha$ for members of the set;
- Adding whole numbers mod $\ell$ using clock arithmetic to always get an answer in the range 0 up to $\ell-1$. I'll write $\mathcal L$ for the set and use regular lower case Roman letters such as $a$ for members of the set. I may also multiply members of $\mathcal L$ by writing them side-by-side;
- Adding points on an elliptic curve with $\ell$ points. I'll write $E$ for the set and use upper case Roman letters such as $G$ or $P$ for members of the set. I'll also write $2P$ for $P+P$ and $aP$ for $\underbrace{P+P+\ldots+P}_{a\text{ terms}}$ where $a$ is a member of $\mathcal L$;
- Adding ordered pairs of whole number mod $\ell$ so that $(a,b)+(c,d)=((a+c)\mod\ell,(b+d)\mod\ell)$. I'll write $\mathbf T$ for the set and use bold lower case Roman letters such as $\mathbf x$ for members of the set. I'll also write $2\mathbf x$ for $\mathbf x+\mathbf x$ and $a\mathbf x$ for $\underbrace{\mathbf x+\mathbf x+\ldots+\mathbf x}_{a\text{ terms}}$ where $a$ is a member of $\mathcal L$.
Recovering private keys from ECDSA
In ECDSA, there is an elliptic curve point that everyone knows called $G$. A public keys is a different point $P$ where $P=sG$ where $s$ is a member of $\mathcal L$ known only to the signer. If we can find $s$ given only $G$ and $P$, then we can forge signatures and break ECDSA. The problem of finding $s$ given only $G$ and $P$ is called the discrete logarithm problem. I'll partially describe how a moderately-sized quantum computer might do this.
Mapping from $\mathbf T$ to $E$
Given $G$ and $P$, I'm going to define a function that takes members of $\mathbf T$ and returns members of $E$. If the pair $\mathbf x$ is $(a,b)$ where $a$ and $b$ are in $\mathcal L$, then my function will take $\mathbf x$ to the point $aG-bP$. I'll call this function $F$ so that
$$F(\mathbf x)=F((a,b))=aG-bP.$$
Mathematically, this is quite a nice function for example if $F(\mathbf y)=Q$ and $F(\mathbf z)=R$ then $F(\mathbf y+\mathbf z)=Q+R=F(\mathbf y)+F(\mathbf z)$ (note that "+" is being used in two different ways here), which is a property know as homomorphism. For the purposes of solving the discrete logarithm problem, it is also important to notice that if $\mathbf d$ is the pair $(s,1)$ then $F(\mathbf x+\mathbf d)=F(\mathbf x)$ for all pairs $\mathbf x$. This is because
$$F(\mathbf x+\mathbf d)=F(\mathbf x)+F(\mathbf d)=F(\mathbf x)+sG-P=F(\mathbf x).$$
Periodic functions
You may have encountered expressions similar to $F(\mathbf x+\mathbf d)=F(\mathbf x)$ before. For example:
$$\sin(\alpha+2\pi)=\sin(\alpha),$$
for all $\alpha$, where we say that the sine function has period $2\pi$. Transferring the language of the set $\mathbb R$ to the set $\mathbf T$, we might say that the function $F$ has period $\mathbf d$. In our case however, we can evaluate $F$, but we cannot write down $\mathbf d$ (which is our goal in solving the discrete logarithm problem). This shows that our discrete logarithm problem is related to a problem known as period finding.
Period finding and Fourier transforms
There is a very elegant mathematical theory of real periodic functions called the Fourier transform, which in one interpretation says that all sensible, real periodic functions are just sums of multiples of sine functions and that the multipliers can be systematically calculated. We often deal with functions where one (or two) multiplier(s) of a particular sine function(s) are large and all the other multipliers are small. The large multipliers correspond to signal and the small multipliers to noise in the reception of radio waves for example. The Fourier transform gives a very efficient way of separating signal from noise in real functions. There a similar idea called the discrete Fourier transform for our set $\mathbf T$, that will identify our period $\mathbf d$ as the signal part of our function. The problem is that there are a lot of intermediate calculations that are very similar, but take a lot of time to compute and write down even though we only want a simple answer at the end. Computing this simple answer using a Fourier transform on a classical computer therefore takes too much time and space.
Quantum superposition
A lot of inaccurate statements are made about how quantum computers work, so be wary about how this next section is described. Notionally, a quantum computer can exist in a spectrum of possible states that allow us to think of all the intermediate calculations of a Fourier transform being represented in a superposition. Provided that we never try and explicitly write down ("measure") one of the intermediate calculations, a quantum computer can work through this complex intermediate state get to a final answer where the signal and noise components of the function are in superposition. When we stop and ask the quantum computer to write down ("measure") this answer, it is very highly likely that the computer will write down the signal part that we want. In our problem this "signal" should be a period of the function $F$.
...except
There's a small problem in that $\mathbf d$ is not the only period of $F$. It is also true that $F(\mathbf x+2\mathbf d)=F(\mathbf x)$ for all $\mathbf x$. This means our quantum computer might return the answer $\mathbf d=(s,1)$, but it might also return the answer $2\mathbf d=(2s\mod\ell,2)$ or $3\mathbf d=(3s\mod\ell,3)$ and so on. Analogously, our sine fucntion can be said to have period $2\pi$ or $4\pi$ or $6\pi$... This is a minor inconvenience if we can perform mod $\ell$ division: when our computer returns a pair of numbers $(u,v)$, we will just divide $u$ by $v$ mod $\ell$ and get $s$. Fortunately we know how to do division mod $\ell$ to get whole number answers using the extended Euclidean algorithm.
Finally
That is essentially how a quantum computer might break ECDSA (omitting the details of Fourier transforms, which are probably too lengthy to fully describe here). What does all this have to do with factoring into primes? Again omitting some details, if we wish to know the prime factors of a number $N$, then it almost certainly suffices to find a whole number $2D$ such that $2^{x+2D}\mod N=2^x\mod N$ for all integers $x$. This is because we would then have $(2^D+1)(2^D-1)=2^{2D}-1\equiv 0\pmod N$ and roughly half of the prime factors of $N$ will divide the first bracket and roughly half will divide the second bracket. Euclid's algorithm then would let us split out the primes, but only if we can find $D$. If we now consider the function from all of the integers to the integers mod $N$ given by $G(x)=2^x$, we have $G(x+2D)=G(x)$ for all $x$ and we again have a period-finding problem (except that now our set is all of the integers rather than the integers mod $\ell$). Again, this is solvable with the discrete Fourier transform (though not very easily on a classical computer) and again a quantum computer can handle the large intermediate information in superposition before returning $2D$ as the "signal".
