I believe it's useful to remember the first law of thermodynamics, $dE = dQ + dW$, which tells us that heat Q and work W are both forms of energy (you can increase the internal energy E of you hands by either rubbing them or place them near a source of heat). Energy conservation is its interpretation; if heat $dQ > 0$ is supplied to the system then the maximum amount of work the system would do on the environment is $dW = dE - dQ$ (where, for work done on the environment by the system, dW is negative).
If 'reversibility is assumed (i.e., if there's no friction, and if the heat transfers are done infinitesimally), then we can say $dW = dE - TdS$, otherwise $dW \ge dE - TdS$. Since $F=E-TS$, then $dF=dE-(TdS+SdT)$, and for isothermal processes we have $dF=dE-TdS$ (since $dT=0$). Hence $dF \le dW$.
If the system was a gas at pressure p, enclosed in a container with volume V connected to a piston, upon heating it would expand; it would do work $dW=-pdV$ on the environment. Hence $dF \le -pdV$, and so $\Delta F \le -\int_a^b p \,dV $, where $b>a$. Negative changes in F indicate work done by the system on the surroundings.
In other words, for systems that are in thermal contact with their surroundings (so that the processes that may occur are isothermal), you can think of $\Delta F$ as the maximum amount of work that you'll be able to get out of the system before equilibrium is reached.
If in addition to being in thermal contact the system is 'mechanically isolated', no work can be applied to or extracted from it, and therefore $dF \le 0$ - meaning that equilibrium is reached when F is minimized and is a constant. An example of such as system could be an idealized spin-$\frac{1}{2}$ paramagnet in the presence of a magnetic field, with non-interacting spins. Since $F=E-TS$, for such a system to reach equilibrium, it will find the right balance between between internal energy E and entropy S at the given temperature T, in order to minimize F. If temperature is low, the system will minimize the internal energy by lining the spins in the direction of the field, even though this would lead to high entropy. If temperature is high, it would randomize the spins to increase its entrope, even though this increases its internal energy.
I hope these examples help you form an intuitive understanding for the Helmholtz function for the cases of systems at constant temperature. Other thermodynamic potentials become relevant for other constraints (e.g., constant pressure, etc) and it's possible to think of intuitive interpretations for them as well. I recommend the book 'Conccepts in Thermal Physics' by Blundell.
