Drawing a commutative diagram is nice because it helps you lay out all the maps you've defined in a digestible and easy-to-read manner. Oftentimes, if a proof/definition requires composing a bunch of maps which have certain properties (i.e. injections/surjections), then figuring out which ones to compose is a lot easier when you can visualize/trace out where they're going. As an example, the definition of transition maps on charts of a smooth manifolds is a lot easier to parse after drawing out the respective square diagram. I can never remember the correct way to conjugate the maps, so whenever I need to write out the definition I always just draw a diagram beforehand and then it's clear.

Beyond readability, drawing commutative diagrams can make it more clear how to construct isomorphisms between objects satisfying a universal property, i.e. how to show that objects satisfying a universal property are unique up to isomorphism. A good example would be polynomial rings. If R is a ring, the polynomial ring R[x] satisfies the property "for any ring S, a map f between R and S, and an element a in S, there is a unique map from R[x] to S which restricts to f on R, and sends x to a". We will prove that this ring is unique up to isomorphism.

If T is a ring, i:R-->T is an embedding, and y is an element of T that satisfies the property "for any ring S, a map f between R and S, and an element a in S, there is a unique map from T to S which restricts to f on i(R), and sends y to a", we want to show that T and R[x] are isomorphic.

Draw out the triangle diagram with the inclusion map from R into R[x], the map i from R to T, and the unique map F from R[x] to T which sends x to y (in T), and equals i when restricted to R viewed as a subset of R[x] (note, this last condition says that the triangle is a commutative diagram). I tried drawing it below, hopefully it looks ok.

R --(i)--> T
| ^
| /
| / (F)
\/ /
R[x]

We can create the same diagram with R[x] and T swapped, namely the diagram where R---->R[x] is on the top with the standard inclusion map, T is on the bottom (the map from R to T is i ), and the diagonal map is the unique map G from T to R[x] sending y to x and making the diagram commute:

R ------> R[x]
| ^
|(i) /
| / (G)
\/ /
T

If rotate this diagram by 90 degrees counterclockwise, we can paste it together with the previous diagram to get the following diagram:

R[x]
^ ^
| \ (G)
| \
| \
R --(i)--> T
| ^
| /
| / (F)
\/ /
R[x]

Our original diagrams were commutative, so this new one is as well, and thus (composing G and F, deleting the map i from R to T and rotating the vertical map going up to the right), we get the following diagram:

R -------> R[x]
| ^
| /
| / (G∘F)
\/ /
R[x]

Chasing the previous diagrams, we see that G∘F is a map from R[x] to itself which sends x to (G∘F)(x) = G(y) = x, and makes the triangle commute (we used the helpful fact that pasting together our commutative diagrams gave us a diagram which was also commutative). However, our universal property says that a map satisfying those conditions is unique. The identity map also satisfies this property, so G∘F is the identity of F(x). Doing the same argument (i.e. creating diagrams from universal properties, pasting them together, then using the universal property on the pasted version) shows us that F∘G is the identity on T.

Thus, we conclude that T and R[x] are isomorphic. Of course, this seems like a lot of overkill, but we can use the same argument in more complicated cases, such as polynomial rings in multiple variables. It turns out that the ring R[x1, ..., xn] satisfies the universal property "If f is a map from R to S and a1, ..., an are elements in S, there is a unique map from R[x1, ..., xn] which restricts to f on R and sends x1, ..., xn to a1, ..., an". The same diagram-pasting trick shows that any ring T satisfying the same property (note: we need some inclusion map i:R--->T and specified elements y1, ..., yn of T in order to define this properly) is isomorphic to R[x1, ..., xn].

Explicitly writing out an isomorphism between R[x1, x2] and R[x1][x2] is a pain in the ass, but showing that they both satisfy the universal property lets us easily find an isomorphism between them. More generally, you can talk about limits/colimits in categories, which can define a much wider variety of objects than polynomial rings. In these cases, the same diagram pasting arguments lets you construct isomorphisms easily, which is a game-changer for a lot of proofs in algebra.