Dynamic optimization with assets as state variable: interpreting capital gains and losses
Firstly, the $\lambda$ is by definition the shadow price of the wealth/capital. It is the price when you increase one unit of additional capital. $\lambda$ measures the rate of increase of the maximum value of the utility function as the constraint is relaxed. So, the dynamic equation of $\lambda$ is the change on this variable, which intuitively is the capital gain.

What do you mean by $(a_{t},H_{t})$ diagram ? Do you mean a phase diagram on $(a_{t},c_{t})$ ? 

For your last question, you can realize on your constraint that the consumption increases at the expense of capital gain (c.f $\dot{a}$.) So, this means that every unit of consumption instead of accumulation is a somewhat loss in capital accumulation.