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Quivers weigh one pound and hold twenty arrows. Twenty arrows weigh one pound.

What's the point of having a case that weighs the same as what it holds? It won't let you hold more?

They answer the question of “where did you put your arrows such that they’re easily accessible?” If you tell me, as DM, that you have 20 arrows but no quiver, then I’m wondering where the arrows are and how you’re grabbing them to shoot your bow. Some alternative answers might be fine, but if they’re just stowed in your pack somewhere, then I will probably say you can’t easily shoot your bow as quickly as you otherwise could because you have to go and get out an arrow every time. And the rules back me up there:

Drawing the ammunition from a quiver, case, or other container is part of the attack (you need a free hand to load a one-handed weapon).

With a quiver, we know you don’t need any extra special effort to get your ammunition. Without a quiver, well, that’s unclear and it’s going to be up to the DM to decide.

Also, having stats for a quiver is useful simply because archers usually have one, and people will want to know how many arrows they can hold, how much they weigh, and how much they cost. The Player’s Handbook description answers those questions for us.

Quivers have the same point as a suitcase when you fly on an airplane.

When you check in your baggage at the airport, your suitcase (which has an empty weight of a few pounds) is also filled with clothes and other things that each have their own weight. On the scale at the airport, the result is the total weight of container and its contents. The person at the airport weighs them to make sure that you are not exceeding the weight allowance for luggage.

Total weight = the container's weight plus the contents' weight.

Your quiver, like the suitcase, is a larger thing that carries a number of smaller things. It has an empty weight (1 pound) and a certain volume - the volume is "20 arrows" when full. The contents (arrows) plus the quiver (container) each weigh one pound, which adds up to 2 pounds total weight when full.

For whatever game mechanics reasons they had, the game designers assigned "1 pound" as the weight of a quiver. They could just as easily made it "half a pound" or "two pounds." Note that IRL, quivers also play a role in protecting the arrows from damage, but that's more granular / simulationist detail than we need to get to for this question.

Regarding your actual question, having to do with encumbrance.

From our comments under @KRyan's answer:

@NathanSantee Is your DM using the variant Encumbrance rules? (Basic
Rules, page 60). The basic carrying capacity is 15 times your
strength. – KorvinStarmast 5 mins ago

Yes but I interpreted it as 5 x strength + containers. – Nathan Santee 4 mins ago

Then you read something into the rules that is not there. From the Basic Rules, page 60; PHB p. 176).

Lifting and Carrying

Your Strength score determines the amount of weight you can bear. The following terms define what you can lift or carry.emphasis mine

Carrying Capacity. Your carrying capacity is your Strength score multiplied by 15. This is the weight (in pounds) that you can carry, which is high enough that most characters don’t usually have to worry about it.

Push, Drag, or Lift. You can push, drag, or lift a weight in pounds up to twice your carrying capacity (or 30 times your Strength score). While pushing or dragging weight in excess of your carrying capacity, your speed drops to 5 feet.

[...]

Variant: Encumbrance

When you use this variant, ignore the Strength column of the Armor table in chapter 5. If you carry weight in excess of 5 times your Strength score, you are encumbered, which means your speed drops by 10 feet. If you carry weight in excess of 10 times your Strength score, up to your maximum carrying capacity, you are instead heavily encumbered, which means your speed drops by 20 feet and you have disadvantage on ability checks, attack rolls, and saving throws that use Strength, Dexterity, or Constitution.

You can carry more, but it slows you down

You will note that nowhere in the rules above are the weights of the containers set to zero, nor are they waived. The key result of carrying more than 55 pounds with a strength of 11 is that your speed is reduced by 10. For example, if you are a human, your speed will be 20 instead of 30 until you reduce the total weight to 55 pounds or less.

You asked in a comment why you need containers.

Just as in real life, you typically use containers to hold things, which frees your hands to do what you are trying to do until you need those things in the container.

@Graham points out an exception to the general case:

If you are in a static defense situation tactically (you don't need to move at all) then you may well have a small forest of arrows stuck in the ground in front of you, as you wait for the enemy to attack. Access to them is "reach down, grab one".