Ineed
what is the missing reason in the proof

There's a lot going on, but the main thing to focus on really is the equation 
AE+EB = CF+FD

and on the equation 
EB = CF

Because the second equation (shown above) is true, we can effectively replace the "EB" with "CF" like so
AE+EB = CF+FD
AE+CF = CF+FD

Then we can subtract CF from both sides
AE+CF = CF+FD
AE+CF-CF = CF+FD-CF
AE = FD

When I subtracted CF from both sides, I used the Subtraction Property of Equality, which is the answer needed to go into the blank. 

4 times

Step-by-step explanation:

1 quart is equivalent to 32 fluid ounces.

If each eruption consumes 8 fluid ounces of vinegar. Assuming that the amount of vinegar required per eruption is constant, the number of times Tanvi can make her volcano erupt is:

Tanvi can make her volcano erupt 4 times  with 1 quart of vinegar.