How should I use world maps in my campaign?

I see so many people talking about using maps for their tabletop campaigns, and I want to try it out too. However, after searching online, I haven't found anything that fully explains how to use them. How does world exploration actually work? Do players just say, "I want to go to this forest," and you pull out a forest map where they can do whatever they want while you throw a monster encounter or two at them?

This brings up another major question I have: how does movement work? When they want to travel to a forest from a faraway place, do I wait 5 to 30 minutes in real time for them to get there? Do I manually drag a little cart miniature along the roads step-by-step until I trigger a random ambush, or do they fast-travel out of game and instead of the forest i put them down on a random road while I say something like, "As you were traveling toward the forest, out of nowhere arrows suddenly fly past..."? Is there a recommended ratio for distance versus real-world time, or is that even necessary?

Anything about this would be really helpful. Thank you

I'm somewhat confused as to the actual question here.

I use a large erasable grid map for my combats, and wet erase markers to draw out the terrain in simple details. It just helps me to orient things so I don't have to keep an extra thing in my head; I already do a lot of my descriptions and things off the top of the dome or with a few words of reminder, because that's how I enjoy DMing, instead of pre-written descriptions. 

.When my players travel, I usually give out a regional map that I gave one of my players, point out the hooks they have to each location, and ask where they're going. I don't have much of a scale or anything, and use a few pregenerated encounters on set days, because I find random encounters tedious busywork. Dragging along a little cart in proportionate time is probably an unnecessary measure, and rolling for every hex they enter should only be something you do in hex crawls, most of the time. When my players are actually exploring a place, I have a table of random encounters that I roll for that each hold some kind of details about whatever is going on in said location. I roll to see if there will be an encounter each day, and if so, what encounter they get. It's best to just sum each day of travel up into short descriptions. 

My players use the world map/regional map to decide where they want to go. One person acts as their "cartographer", and has marks on the map to indicate places of interest they've heard of/come across. Standard travel time is 24mi/8hr day, so I'll use the map scale to determine how many days it will take for them to get wherever they're going. I only use maps with miniatures for combat. Exploration is done entirely with descriptions/theater of the mind. I have a wet-erase grid/scatter terrain for random/unexpected encounters, and then I get really into 3d terrain building for planned/important combats. 

If they're traveling on a well-used road I just fast-forward through travel time, unless there's something that I want to happen from a plot perspective. There's no reason to play through every travel day on the road, I'll just describe the terrain changes if they're going through different climates or something. I don't think dragging a cart on the table or having IRL pauses in time to represent travel is really necessary. I just describe the passage of time, and if they want to interrupt me to role-play or do something specific then I let them. 

 If they're traveling through dangerous/wild territory, I'll use encounter tables to describe environmentally-relevant things they could come across. Usually not combat, sometimes just bad weather, other times challenges they have to deal with to keep moving forward (how do you want to cross the giant crevice) or risk losing time, or NPCs/social encounters (you see a hunter hiding in the brush, bow aimed for a deer.). And sometimes they just get "your travel through this stretch is uneventful" and I'll make up some descriptions of the terrain around them. If they're in a place where combat is likely, I'll have it pre-planned and put it in when it makes sense. I hate throwing together combat encounters on the fly, it just feels tedious and annoying.  

If they're actually exploring a place, like doing a deep exploration of a forest looking for something, then I have much more intricate random encounter tables and/or pre-planned encounters for each area. Like if they're traveling through a forest looking for some evil coming from ancient ruins, I'll have encounters related to that as they get closer, and then detailed descriptions and/or encounters for each "room"/area in the ruins. Not every room needs combat though, so I won't necessarily have a map for each individual room in something like this.