It is really not that bizarre or abstract.
Shares in a company represent fractions of ownership of a company. If you think GE as a company is worth X billion dollars, and there are Y million shares outstanding, then you think X / Y is what each share is worth.
Owning shares also gives you the right to vote (along with the other shareholders) on how the company is run, kick out bad management, and so forth.
That is what they say but people break it down very technically very quickly to lose the overall point. If you buy shares of GE, you do own a percentage of the company on paper. However, you can’t actually point your finger at any concrete thing you own except for the (virtual) certificate. You can vote to help make decisions but your vote counts for very little for any reasonable amount of stock and it is perfectly possible for it to count for zero among all outside shareholders if the company itself still owns a controlling number of shares and can override anything.
As a shareholder, the direct goal isn’t to make sure the company is as well run as possible at all times. The direct goal is to get the share price up high enough so that you can pass your shares onto another person and take your profit. Repeat. Stock prices don’t go up or down just based on company performance. They are also influenced by investment cycles and speculative bubbles. They can go down sharply even if nothing has changed fundamentally just based on fear and panic of other investors. Voting to ensure long-term viability isn’t always the goal. You may hope they will push out a new product as fast as possible to get the share price up and then get rid of it taking the profit with you just based on other people’s guesses and herd mentality. Pricing stocks would be easy if it were just simple math problems but it doesn’t work that way.
I am not saying the market doesn’t move based on real financial prospects for the company because it tends to but it isn’t a direct, lock-step relationship.