(Please see each class, spell, or weapon for a full description of damage. Below is a simplified, generalized breakdown of damage dealing in 5e.)
A lot of new players to Dungeons and Dragons (fifth edition) are confused about how much damage their character can do at higher levels, particularly with melee or ranged weapons (swords, daggers, bows, crossbows, etc). Let’s explain how damage works above level 1, both for casters and non-magic weapons.
A lot of players get super confused when they see “Hit Dice” in their character sheet, and assume that means they roll their hit dice to do damage. This is poorly worded on the character sheet, as well as in the official books. The “Hit Dice” refers to the amount of HEALING you receive during a short rest. For instance, as a level 5 druid, your hit dice would be 5d8 – when you do a short rest, you’d roll 5d8 and recover that many hit points. If you complete a long rest, you recover all your hit points, and all your hit dice are available to be used again. It has absolutely nothing whatsoever to do with how much damage your character does. Maybe it’s easier to think of “Hit Dice” as “Health Recovery” instead.
Melee / Ranged Weapons
First thing to remember is that the base damage of a melee or ranged weapon does not increase with level. For example, a dagger does 1d4 piercing damage at character level 1. At character level 8, a dagger will still do 1d4 piercing damage. This may seem like a ridiculously small amount of damage for a level 8 character, and it is - except you are not likely attacking with just one dagger, and you'll also be using things like "multi attack" and "sneak attack" and "surprise attack", etc. These attacks come available as you level, along with others. So, attacking with daggers at level 8 (let's assume you are using two daggers, as a rogue), you would be doing 1d4 + proficiency modifier per dagger, and if you're attacking from the shadows or something similar, you'd also get your sneak attack damage (add a number of d6 rolls for damage, depending on level) for each of your turns. As you can see, this would increased the base damage of the 1d4 dagger by leaps and bounds. Most classes have extra attacks, surprise attacks, etc. Add to this any magic weapon damage, such as a +3 dagger (does an additional +3 per hit) and you can see how effective - and deadly - these weapons can be.
Magic / Spells
Magic users have always-available spells called Cantrips. These level zero spells do not use a spell slot, and can be cast at will. Some of the cantrips are damage dealing spells, some are healing spells, illusions, etc. There are many cantrips to choose from for your magic user character. Cantrips typically do a base amount of damage or healing, and can advance as your character levels. Let's look at the cantrip "Acid Splash" as taken from the Player's Handbook: You hurl a bubble of acid. Choose one creature within range, or choose two creatures within range that are within 5 feet of each other. A target must succeed on a Dexterity saving throw or take 1d6 acid damage. This spell's damage increases by 1d6 when you reach 5th level (2d6), 11th level (3d6), and 17th level (4d6). This is a base amount of damage, but it doesn't use a spell slot to cast it. Cantrips are "freebie" spells. You can cast them at will, provided you know that cantrip. Damaging spells, such as the third level spell "Fireball" for instance, do a LOT more damage, but require you to use a spell slot to cast, as well as having it prepared. Let's take a look at the "Fireball" spell description from the Player's Handbook: A bright streak flashes from your pointing finger to a point you choose within range and then blossoms with a low roar into an explosion of flame. Each creature in a 20-foot-radius sphere centered on that point must make a dexterity saving throw. A target takes 8d6 fire damage on a failed save, or half as much damage on a successful one. The fire spreads around corners. It ignites flammable objects in the area that aren't being worn or carried. At Higher Levels. When you cast this spell using a spell slot of 4th level or higher, the damage increases by 1D6 for each slot level above 3rd. 8d6 base damage... That's pretty hardcore damage, and that damage increases as you level and cast it using a higher level spell slot.
It should be clear now how damage works in 5th Edition of Dungeons and Dragons. Weapon damage doesn’t typically “scale up” with your levels, but you do gain extra hits, attacks, and very likely magic bonuses that increase the damage your weapons will do.