Reading Binomial Tables While reading a table of cumulative binomial probabilities, if I need to find the probability of, for example, exactly 4 successful events happening and all the rest failures occurring, how would I go about that ?
The table you show is for when $n=20$ and is cumulative.  If you are interested in finding the probability of at most $4$ successful events and the rest failures... you look at the row which begins with the bold 4, then look across until reaching the column which begins with the bold 0.xx where 0.xx is the probability of success.  Trace your finger across the row and down the column to see where they meet.  For example, at most $4$ successes out of $20$ where probability of success is $0.25$ occurs with probability approximately $0.415$.



Since you are interested in the value for exactly $4$ successes rather than at most $4$ successes, you can subtract the value right above it.  I.e. $Pr(X= 4) = Pr(X\leq 4) - Pr(X\leq 3)$, which for the case where $p=0.25$ yields approximately $0.415 - 0.225 = 0.19$.

Doing this directly yields $\binom{20}{4}0.25^40.75^{16} \approx 0.189685\cdots$ confirming the result above.

If you were interested in probabilities where $n$ were some number different than $20$ then you will need a different table.  If you are in a scenario where you have a calculator or calculator software available, I see no reason to bother with tables in the first place and would instead use the usual formulas directly.
Go into EXCEL, in the function field enter BINOMRASP and calculate. In this case, type =BINOMRASP(4;20;0.25;FALSE)