Lattice Reductions in Lattice Crypto
Well to start off, lattice reductions are of interest in the context of lattice crypto because, as already stated in other answers, they allow us to find short vectors in a lattice, which relates to SVP (finding the shortest vector in a lattice).
We care about solving SVP because it is believed that there may be a security reduction from Ring Learning With Errors (RLWE) to SVP. So if we can solve SVP (via some technique e.g. lattice reduction) we may be able to break schemes based on RLWE e.g. these homomorphic encryption and key exchange schemes.
It should be noted that reduction techniques such as LLL do not allow us to solve SVP efficiently for lattices in a high dimension and as such they are not effective for breaking these schemes.
Lattice Reductions in Number Theoretic Crypto
But there's also an application to "classical" number theoretic crypto. I'll explain one (of several) possible applications. Suppose we want to attack (EC)DSA where nonces are generated with biased bits. It turns out we can use a lattice reduction to recover the private signing key.
If we can get enough message/signature pairs that all used nonces with the same bias then we can actually write a system of equations (with each equation corresponding to a message/signature pair) that can be interpreted as a lattice (just like how you can might use an augmented matrix in linear algebra to represent a system of equations).
We can then actually perform a lattice reduction (e.g. LLL) on this system of equations. If we have sufficient message/signature pairs then with high probability one of the entries in the reduced basis will be the private signing key. The details of this attack can be found in this paper.
For more information on lattice reductions in this context I'd encourage you to look at the paper Lattice Reduction: a Toolbox for the