MPI stands for Multiple Precision Integer. Multiple precision arithmetic is what you need when you work with integer types that go beyond the machine width $w$.
The basic idea is simple, you represent a large integer with multiple fixed-width words where the i-th word is the i-th "digit" in base B where $B = 2^w$.
For example, most current machines are 64-bit so the width $w$ is 64, so with a single word you can represent unsigned integers up $2^{64}-1$. To represent integers larger than $2^{64}-1$, say a 1024-bit integer as in your RSA example, you use $\lceil{1024 / 64}\rceil = 16$ words $a_0, a_1, a_2, \ldots, a_{15}$. Then your integer $x$ of choice is encoded as
$$
x = a_0 + 2^{64} a_1 + 2^{2*64} a_2 + \ldots + 2^{15*64} a_{15}.
$$
Note that this is essentially a $1024$-bit representation, the only difference is that the bits are grouped into blocks of size 64.
Operations like additions, multiplication etcetera are implemented by building on machine addition, multiplication and so on, but of course additional work is needed to take care of carries and the like. I am not sure what the Linux kernel is using, but in the GNU/Linux world a widely used multiple precision arithmetic library is the GMP.
