First of all, by Morse theory you know that $\mathcal{M}^+$ is obtained by attaching an $m$-cell to $\mathcal{M}^-$, i.e. $\mathcal{M}^+=\mathcal{M}^-\cup_{\phi}e$, where $e=D^m$ is an $m$-cell and $\phi:\partial(e)=S^{m-1}\to \mathcal{M}^-$ is some attaching map. Therefore you can compute $H^{\*}(\mathcal{M}^+,\mathcal{M}^-)$ by excision (you excise $\mathcal{M}^-\setminus\phi(S^{m-1})$), and get that $H^{\*}(\mathcal{M}^+,\mathcal{M}^-)\simeq H^{\*}(D^m,S^{m-1})$ that has exactly one nonzero cohomology in degree $m$.
As for the second question, I think it's more intuitive looking at homology (it's the same since we are looking at coefficients in a field). There you have the sequence
$0\to H_m(\mathcal{M}^-)\to H_m(\mathcal{M}^+)\to H_m(\mathcal{M}^+,\mathcal{M}^-)\stackrel{\partial}{\to} H_{m-1}(\mathcal{M}^-)\to M_{m-1}(\mathcal{M}^+)\to 0$.
Moreover, the generator of the unique nonzero class of $H_m(\mathcal{M}^+,\mathcal{M}^-)$ can be thought of the fundamental class $[e]$ of the cell you attached, and $\partial[e]=\phi_*[\partial e]$. In other words, the boundary operator sends $[e]$ essentially to the class of its topological boundary. If $\partial[e]=0$, it means that $\phi_{\*}[\partial e]$ is a boundary in $\mathcal{M}^-$, i.e. there exists an $m$-cycle $e_-$ entirely contained in $\mathcal{M}^-$ such that $\partial[e]=\partial[e_-]$. In particular $\partial[e-e_-]=0$ and you can say that $e$ can be completed to a cycle, contained in $\mathcal{M}^-$ away from the critical point}.
Think of the usual example of the torus, with the morse function being the heigth $z$. Let $p_1$ be the second critical point (the first critical point of index one) with critical value $z_1$, let $\mathcal{M}^-=z^{-1}([0,z_1-\epsilon])$ and $\mathcal{M}^+=z^{-1}([0,z_1+\epsilon])$. Then $\mathcal{M}^+$ is obtained by attaching a $1$-cell $e$, that is a little segment passing through $p_1$ and "going down". Its boundary $\partial[e]$ is just two points in $\mathcal{M}^-$. You can check that $\partial:H_{1}(\mathcal{M}^+,\mathcal{M}^-)\to H_0(\mathcal{M}^-)$ is zero, and in fact $\partial[e]$ is the boundary of an arc $e_-$ entirely contained in $\mathcal{M^-}$ (for example the arc through the minimum point). it's quite clear that we just "completed" $[e]$ to a cycle $[e]-[e_-]$ (that you can see as a closed nontrivial closed curve in the torus) that is "almost entirely contained in $\mathcal{M}^-$ away from the critical point.
If $\partial\neq 0$, this means that we are killing some cohomology, that we created at some point before. Take for example the sphere $S^2$, embedded in $\mathbb{R}^3$ in the shape of a "$\bigcap$", and again with $z$ as a Morse function (just wiggle it a bit so that the two minima have different values). Then from bottom-up, you first see two index $0$ points (the two minima), that give you two 0-cells. Then, as you go to the third critical point (the first critical point of index 1), you notice that $\partial \neq 0$ and so the contribution of that critical point is to kill something we created before. In some sense then, this morse function is not "efficient", in the sense that creates too many critical points, that will be killed later anyways. An "efficient" Morse function only creates critical points when they really contribute to the homology of the space. "Efficient" Morse functions are called perfect.
Here is the formal definition of perfect morse function: as you said, every critical point of index $m$, might give rise to at most one new generator of $H_{\*}(\mathcal{M}_+)$. So if $c_m$ denotes the number of critical points of index $m$, we have that $c_m\geq \dim H_{\_*}(M)$, where $M$ is our manifold. The Morse function is called perfect if the inequality above is an equality, for every $m$.
