A arcane focus is a simple object – like a magnifying glass or jeweler’s loupe – that can serve as the tool needed for certain “ritual” spellcasting. This particular kind of wizard, warlock, or other mage prepares spells in advance and they must be cast in conjunction with an arcane focus.
Originally introduced because few fantasy worlds had any reason to do unit conversions between weights of different items which are more often measured by volume than weight when computing their gold piece value according to their weight/volume conversion rate. D&D spells cannot use calculations based upon distance, time, energy output, power input etc., so it was realized this early that the types of spells available might need to be affected by one’s carrying capacity.
As a result the wizard’s portable hole spell, which was initially made available as one of the basic spells for 0th level spellcasters, became a pure liability for anyone who actually needed to use it, since players were completely dependant on both their gold pieces and how many coins they could carry before their weight limit was either reached or exceeded. The same situation happened with the cost of other items, which was originally supposed to be calculated by weight/volume rate but because none of them were ever intended to actually weigh anything this quickly became an unrealistic scenario full of no win situations for D&D players.
Another emerging problem was that it never made any sense why carrying capacity would affect the number of spells that could be cast, since it would only make sense if carrying capacity was meant to make you unable to carry objects like spellbooks or scrolls (which was later covered by making them weightless) but otherwise it did not help at all when actually casting spells.
As a result arcane focuses were introduced in order to create an in-game explanation for why carry capacity had some sort of altered effect on spellcasting, especially since carrying capacity was originally only supposed to be an optional rule which had no impact on play at all. This is why most spells today are still based upon the original assumption that carry weight has zero influence on how many spells can be cast at once.