Edit: Looks like "a few hours" plus "20 minutes" passed, but Yaakov is still unconvinced, so I'm adding the pair warm-up. I'll go from the back, the way I figured it myself.
If you have $m\ge 2$ numbers with sum $S$, then the maximal pair has the sum $\ge 2S/m$.
If you have several marked positions in the matrix so that each column that contains a marked position at all contains at least 2 of them, then the maximal column pair has the sum $\ge 2S/N$ where $S$ is the sum of all numbers in marked positions and $N$ is the number of marked positions.
Thus, if we mark all positions of two largest numbers in each row, and find a submatrix with $a$ rows such that each its row contains two marked positions and each its column contains at least 2, then we will have the maximal column sum at least $2a/(2a)=1$ because $a$ is the sum of marked entries in this submatrix and $2a$ is the number of entries (I again assume $R=1$.
It remains to find such a submatrix. To this end forget all initial numbers and put free variables in the marked positions and $0$ everywhere else. Then we have $2n$ free variables. Restrict them by the equations row sums=column sums=0 ($2n-1$ independent homogeneous equations). We have a non-trivial solution. Look at each row with a non-zero free variable in that solution. It must have another free variable $\ne 0$ too to balance the first one. Now look at the columns with at least one non-zero free variable and observe the same. Thus these rows and columns will give us the desired submatrix.
The case of triples is complicated by the fact that we have to use the extra $n$ free variables and $n$ equations in a somewhat inventive way. To have the analogue of 2) for triples (with $3S/N$), we would need one extra equation for each column and each row to avoid two-entry solutions (any non-trivial solution of $x_1+\dots+x_n=x_1+2x_2+\dots+nx_n=0$ has at least 3 non-zero entries), but that's too much (which isn't surprising because that would prove $C\ge 1$, which is false). So we try to enforce columns, but then we cannot guarantee all three entries in the rows, only 2. If the sum of any 2 of the 3 largest entries in the rows were $\ge 1/2$, i.e., if there were no entries $\ge 1/2$, that would be enough because if in the similarly chosen we had $b$ rows with 2 entries and $c$ rows with 3 entries, we would have $S\ge \frac 12 b+c$ and $N=2b+3c$, resulting in the bound $3\frac{\frac 12 b+c}{2 b+3c}\ge\frac 34$, as required. But we may have entries $\ge 1/2$, so we need to take special care of them. We do it by sacrificing one free variable in each row containing them, which forces us to sacrifice one equation too. Then the columns for which we don't have the additional equation would have to be augmented by a third element, which increases $N$ by their number (hence $N+a$ instead of $N$ in the denominator). The estimate of $S$, however, improves to $\frac 34a+\frac 12b+c$, and we can still manage.
I hope this warm-up facilitates reading a little bit. Apologies for extra misprints if I introduced them :-).