Why do binomial expansions involving surds get closer to integers as they get larger? Suppose I have a binomial expansion of the form:
$$
(2+\sqrt{3})^n
$$
Why is it that as $n$ approaches $\infty$ that the value of the expansion becomes closer and closer to being an integer?

If $\alpha_1$ is an algebraic integer (which $2 + \sqrt{3}$ is) then it's the root of a monic irreducible polynomial $f(x) = x^d + \dots$ with integer coefficients, which here is
$$f(x) = (x - (2 + \sqrt{3}))(x - (2 - \sqrt{3})) = (x - 2)^2 - 3 = x^2 - 4x + 1.$$
This polynomial has some other roots $\alpha_2, \dots \alpha_d$, the conjugates of $\alpha_1$, and then you can show in various ways that:

Claim: The sequence
$$p_n = \sum_{k=1}^d \alpha_k^n$$
of power sums is always an integer.

Here this sequence is $(2 + \sqrt{3})^n + (2 - \sqrt{3})^n$ as Jaap says in the comments. This is easiest to understand in the quadratic case $d = 2$ but it holds more generally.
If it further happens that the other roots $\alpha_2, \dots \alpha_d$ all have absolute value less than $1$, then their contributions to the power sum above decay exponentially as $n \to \infty$, and then for $n$ large enough that the sum of these contributions is less than $\frac{1}{2}$ (which happens quite quickly), $p_n$ will be the closest integer to $\alpha_1^n$. The real algebraic integers with this property are called Pisot-Vijayaraghavan numbers and they're somewhat rare but they do exist. The most famous one is probably the golden ratio $\phi = \frac{1 + \sqrt{5}}{2}$, whose conjugate is the "other" golden ratio $\varphi = \frac{1 - \sqrt{5}}{2}$. The sequence of power sums
$$L_n = \phi^n + \varphi^n$$
is the Lucas numbers, a close cousin of the more famous Fibonacci numbers, and $|\varphi^n| < \frac{1}{2}$ for $n \ge 2$ so we get that

Claim: For $n \ge 2$, $L_n$ is the closest integer to $\phi^n$.

There's an analogous formula for the Fibonacci numbers which goes
$$F_n = \frac{\phi^n - \varphi^n}{\phi - \varphi}$$
and similarly it implies

Claim: For $n \ge 1$, $F_n$ is the closest integer to $\left[ \frac{\phi^n}{\sqrt{5}} \right]$.

$2 + \sqrt{3}$ has this same kind of relationship to the sequence
$$p_n = (2 + \sqrt{3})^n + (2 - \sqrt{3})^n$$
which can (this is one of the ways to prove it always consists of integers) equivalently be defined as the sequence satisfying $p_0 = 2, p_1 = 4$ and the recurrence relation
$$p_{n+2} = 4 p_{n+1} - p_n.$$
This sequence begins $2, 4, 14, 52, \dots$ and I don't think it has a name but it's A003500 in the OEIS.
