Suppose we consider an Ideal gas. (Ah, one more definition, Ideal gas is a gas in which we neglect attraction between particles, therefore, no potential energy, thus here internal energy just refers to total kinetic energy) and make it undergo Isothermal process, then BY DEFINITION, temperature or Average kinetic energy of the molecules remains a constant. If Average kinetic energy is a constant, then the total kinetic energy must also remain a constant. So look at the internal energy of this gas. Since internal energy for an ideal gas is just it’s total kinetic energy, by definition, the internal energy must be a constant. Does that make sense now?

However, if we break that assumption, that particles are not attracted to each other, then potential energy comes into the picture. Now, in an isothermal process, even though temperature (average kinetic energy) is a constant, it’s potential energy can definitely change, thus now it’s internal energy can definitely change.

A practical example would be during phase change. Suppose you consider water boiling. Since water is a liquid, you can definitely NOT neglect the attraction between particles and hence potential energy cannot be neglected at all. Hence during boiling, all the heat energy supplied (or taken up by the liquid) increases the potential energy of the system (makes particles farther and farther away), but keeps the total kinetic energy a constant.

So the temperature definitely remains a constant, thus by definition this is an isothermal process, but since potential energy is increasing (like crazy) the internal energy of the system is definitely increasing.