Algorithm to find maximum number of floors you can check with N eggs and D maximum drops
First we'll clarify the problem a bit. We have $n$ eggs, which we can drop from any floor we want. We also have the constraint that we are allowed a total of $d$ drops of these eggs (rather than $d$ drops per egg). Obviously, when we drop an egg and it breaks, we can't drop it again. Let $f(n,d)$ be the maximum number $M$ such that with $n$ eggs and $d$ drops, we can completely describe the behavior of our experiment for floors $1,\dotsc,M$. 

In other words, with $M=5$ the possible outcomes are [YYYYY] (the eggs break when dropped from all 5 floors), [NYYYY] (the egg survives a drop from floor 1 but no higher floor), and [NNYYY], [NNNYY], [NNNNy], and [NNNNN]. Obviously, in all possible descriptions, all the N's must come before all the possible Y's 

With that out of the way, let's tackle the recursion 
$$
f(n,d)=f(n-1,d-1)+1+f(n,d-1)
$$
Our plan of attack is to drop the first egg from floor $x$, yet to be determined.
That egg either breaks or doesn't:


If it breaks, we know it will break when dropped from any higher floor as well, so we're left with $n-1$ eggs and $d-1$ more drops. In other words, we're looking at the lower floors and there will be $f(n-1,d-1)$ of them, namely floors $1, 2,\dotsc, f(n-1,d-1)$.
Since we've already settled the behavior at floor $x$, we need to have $x=f(n-1,d-1)+1$.
If it doesn't break, we know it won't break when dropped from a lower floor, so we still have $n$ eggs and $d-1$ drops remaining, which will mean that we can determine the behavior of the higher $f(n,d-1)$ floors. They will start at floor $x+1$ and will cover floors $x+1, \dotsc, x+f(n,d-1)=f(n-1,d-1)+1+f(n,d-1)$, exactly as the recursion stated.


For example, you correctly observed that $f(1,5)=5$ but your value for $f(2,5)$ was low. In fact, $f(2, d)=d(d+1)/2$, which is 15 in the case where $d=5$. Start with the first drop from floor 5, and you'll see that you can indeed determine the egg behavior at all of the first 15 floors.