Gerry Myerson's answer is very good. However, I wanted to offer a different, more simple-minded answer.

To begin with, my answer requires that I show every $K_6$ has two monochromatic triangles. The proof is as follows:

Suppose there is only one monochromatic triangle As you noted, we know that every $K_6$ must have at least one monochromatic triangle. For ease of reference, I call the vertices in this monochromatic triangle $A$, $B$, and $C$; I call the other three vertices $D$, $E$, and $F$. Suppose, without loss of generality, that the triangle formed by $A$, $B$, and $C$ is red. At most one of the edges connecting $D$ to $A$, $B$, and $C$ can be red; otherwise we have must have at least one more red monochromatic triangle. We can reason similarly for $E$ and $F$. Now take $D$ and $E$, and notice that, by the Pigeonhole Principle, there must be at least one vertex of $A$, $B$, and $C$ such that $D$ and $E$ are both connecting by a blue edge to that vertex. If $D$ and $E$ are connected by a blue edge, then we have a monochromatic triangle, so they must be connected by a red edge. We can reason similarly for the edge connecting $D$ and $F$ and the edge connecting $E$ and $F$. But then $D$, $E$, and $F$ must form a red triangle. So we have a contradiction and there must be more than one monochromatic triangle in $K_6$.

Now I turn to the involving $K_7$.

For ease of reference, I refer to the seven vertices in $K_7$ by the letters $A$, $B$, $C$, $D$, $E$, $F$, and $G$. The subgraph of $K_7$ made up of $A$, $B$, $C$, $D$, $E$, and $F$ must have two monochromatic triangles. Without loss of generality, suppose $A$ is a vertex in one of the monochromatic triangles of the subgraph with vertices $A$, $B$, $C$, $D$, $E$, and $F$. Then the subgraph made up of up $B$, $C$, $D$, $E$, $F$, and $G$ has a monochromatic triangle that wasn't in the subgraph made up of $A$, $B$, $C$, $D$, $E$, and $F$ *or* the subgraph made up of $A$, $B$, $C$, $D$, $E$, and $F$ had at least three monochromatic triangles at least two of which are in the subgraph consisting of $B$, $C$, $D$, $E$, and $F$. Either way, this means that $K_7$ has at least three monochromatic triangles. If $K_7$ has three monochromatic triangles, then by Pigeonhole Principle at least one vertex must be in two monochromatic triangles, which means the subgraph of $K_7$ consisting the six vertices other than this vertex must have at least two monochromatic triangles. So $K_7$ must have at least four monochromatic triangles.