"Weights" in LLMs are the "final answer" numbers from the results of model training, and these are the engine of the LLM model. Lets use chocolate chip cookies as an analogy.

The chocolate chip cookies are produced with ingredients, a recipe, labor effort to produce the dough, and then cooking energy/effort to bake chocolate chip cookies. In a traditional AI company, the company gets the ingredients, they write their own recipe, do all the dough creation, and then the energy/labor for baking, and charge you money to get chocolate chip cookies.

An open-weights AI company got all the ingredients, wrote their own recipe, labor effort to produce the dough, and instead of charging you for the dough, you get as much uncooked dough as you want forever. Your only task is to take the dough and cook it yourself and you've got free chocolate chip cookies. You can make as many cookies as you want with your own oven. However, you are not given free raw ingredients, nor are your given the recipe to alter it in a way you might like. You can only get the dough for free.

So open-weight AI models (uncooked dough) are LLMs you can use on your own computers (oven) for free and have LLM output ( chocolate chip cookies), but you don't get the training data (ingredients) nor the training parameters (recipe) that built the open-weight model.