ELI5 - What do (prop) traders do

You are conflating a few different businesses here because "prop" trading is such an ambiguous term. I'm not an expert on S&T and the prop world but this may help:

Pre-08 the banks and S&T desks could take on a lot more risk, and they even had dedicated proprietary trading desks who were mostly taking risk positions on behalf of the bank itself. These desks actually spawned lots of hedge fund guys cus it was relatively similar back then (see goldman risk arb desk). These were shut down following the financial crisis. The main business is supposed to be acting as a broker dealer / market maker where your customers are institutions looking to buy and sell securities. In that capacity it opens up opportunities to take on more directional bets where you look to get filled on trades so you can take that directional bet rather than purely fulfilling client orders and profiting off the bid ask spread. Following the financial crisis, banks were no longer allowed to have dedicated prop desks, and there are strict restrictions on how much exposure they can take and how much capital they can direct to these businesses. Also add on increased liquidity and transparency in a lot of markets + algo driven executions, and the space has shrunk from what it used to be. Despite that, FICC desks still do some of these activities at the banks today, but again its market making first and tight restrictions. 

HFs are pools of capital from outside investors looking to make a return on their money. Don't really need to explain the HF world here.

Dedicated prop firms range the entire spectrum today. They are (usually) focused on acting as a market maker and are using their own internal capital (they don't have investors, not considered an investment / part of an investor's asset allocation). The strategy can be similar to some hedge funds, and also be wildly different from a ton of funds - depends on the HF you are comparing it to. Also prop firms are usually not holding positions for as long of a time, with bets / positions being mostly intraday. You can have very algo oriented places like Jane Street, and you can also have more discretionary type places. There are also total chop shops out there where you have to bring your own capital and get to use the firm's platform, access to flow, etc., and then the firm keeps a portion of your gains. 

I guess this is redundant now, but the biggest difference is that prop firms are primarily market makers, leveraging the firm's own capital, fulfilling market orders, while hedge funds are pools of external capital, looking to generate returns for their investors, with much wider flexibility and diverse strategies. I'm willing to bet there are PMs at millennium these days who are doing the exact same thing as some prop guys because the sharpe is good enough in a market neutral environment but I don't know the granular details of those strategies enough to offer much insight.