Calculus of Variations can seem a little daunting, however hopefully a few comments will make things seem a little less scary.
1) You can think of the term "functional" in this case as something that sends functions to real numbers. The definition of a functional is actually a little more technical than that (it is a map from a vector space to its underlying scalar field - see Functional (mathematics) - Wikipedia, the free encyclopedia) - in your case the vector space is the space of continuously differentiable functions with and and the "field" is the real numbers. However, thinking of a functional as something whose domain is a certain set of functions (in your case continuously differentiable functions with and ) and whose range is the real numbers is good enough for now.
2) You have actually done most of the hard work already by solving the Euler-Lagrange equation to find a possible minimizer. I will denote your possible minimizer by i.e. Note that my notation is a little dangerous because we haven't actually proved that is actually a minimizer of yet. The last thing we need to note is that the condition was left out in the original post where you mentioned The condition is essential; represents a small pertubation of , however we don't perturb the endpoints of , which is where the condition comes from.
Now we compute
expanding this we get
If we integrate by parts on the second term on the RHS we obtain
Now use and on the previous line to get
Using the last line in we have
This proves that really is a minimizer of for the class of functions that you're considering.
Does this help? Let me know if anything is unclear.