I am just now beginning my study of electrodynamics, and having a bit of trouble understanding the concept of giving energy and momentum to fields (specially momentum). I understand that the fields have energy and momentum in the sense that they can "offer" it to charged particles when they interact with the field; in other words, we define it as so so we can keep talking about conservation of energy and momentum like we do in other types of dynamics. But is there any other meaning to saying fields can have energy, momentum, etc other physical properties like mass (because I heard some fields can have that as well)? Furthermore, how should I conceptualize and visualize fields having those properties, for example, having momentum?
Fields can exert forces on charged particles.
Force by definition is the rate at which momentum changes.
Meaning when the field exerts a force on charged particles it increases its momentum, this momentum has to come from somewhere, this is why it's important to prescribe momentum to the fields.
This is especially important when there is an inherent time delay to the field. Normally you may just say "well the momentum increase on object A, comes from a decrease of momentum from object B" but because there is a delay to the field, this explanation is insufficient
Perhaps the most direct consequence is radiation pressure. While small in everyday life, it's measurable on Earth with a Nichols radiometer. In space, it's a big deal, with a substantial effect on comet tails, interplanetary trajectories, and telescope pointing.
The TESS mission occasionally sees a rather spectacular demonstration of the effect of radiation pressure in space.
As you mentioned, the energy and momentum stored in the fields can be "given back" to particles. This does make it seem like we defined field energy and momentum just for the sake of it and to keep using energy and momentum conservation, but there is more: they gravitate.
In General Relativity, the presence of energy, momentum, and stresses is what causes gravity. It is a bit more complicated than Newtonian gravity, in which everything comes from energy (which is equivalent to mass by means of $E = mc^2$). The energy, momentum, and stresses stored in the fields do gravitate, and you have, for example, the fact that a neutral particle falling into a black hole with no charge behaves differently than the same neutral particle falling on a charged black hole.
Perhaps another way of noticing how these effects are relevant is by means of the Rindler–Denur capacitor paradox, in which one computes the energy stored in a capacitor in two different reference frames in two different ways. The results ends up being inconsistent unless one takes into account the stresses caused by the field.
What does it mean to give the fields energy and momentum?
The concept of a field in physics
In physics, a field is a physical quantity, represented by a scalar, vector, or tensor, that has a value for each point in space and time
One has to define what is mathematically represented with the term "field". For electrodynamics one has to be clear when one is discussing the pre-Maxwell or the post Maxwell view. Before the brilliant gathering of the laws of of electricity and magnetism into the classical electromagnetic theory. Before, there was no mathematical connection between light and electric and magnetic fields. Lorentz transformations and four vectors are inherent in Maxwell's theory, and it is the four vector that imposes momentum conservation in addition to energy conservation and describes mathematically electromagnetic radiation.
This link to Feynman lectures I think makes it clear that it is the four momentum Lorentz transformation that involve momentum conservation in classical electrodynamics.