0. The basic idea is to show that an algebraic number -- i.e., one that's a root of a polynomial with integer coefficients -- can't be very well approximated by rational numbers ... and then to construct a number that can be very well approximated by rational numbers.
1. Suppose you've got a polynomial p, of degree d, with integer coefficients. Let x be one of its roots, and let a/b be any rational number. How close can it be to x? Well, let's ask a slightly different question: how close can p(a/b) be to 0? Answer: it's always (some integer)/b^d, and unless x actually equals a/b that integer isn't zero, so is >=1 or <=-1; so abs(poly(rational)) >= 1/denominator^degree.
2. OK, so how close can a/b be to x? Well, roughly speaking the ratio between p(a/b)-0 (which we know isn't too small) and a/b-x (which is what we care about) is close to p'(x), provided a/b is close to x. So (after filling in the details) we find that: if x is a root of an integer polynomial of degree d then there's a constant C such that every rational number has abs(rational-x) >= C denominator^d.
3. Now let's construct a number that is better-approximated by rationals than that allows. Specifically, take a decimal that looks like 0.110001000... where the spacing between the 1s grows very fast. The traditional thing is to take 10^-1! + 10^-2! + 10^-3! + ... so let's do that. Then truncating after 10^-k! leaves an error that's approximately 10^-(k+1)!; in other words we have a rational number with denominator N=10^k! and an error on the order of N^(k+1), so for any integer polynomial of degree d, once k gets somewhat larger than d this approximation is "too good", so this number can't be a root of that polynomial. So it can't be the root of any integer polynomial; in other words, it's transcendental.