Most often organizations allow voting for everybody it seems, this site shows a message all the time telling that I don't have enough reputation.
Stack Exchange sites have a gameified system of moderation to encourage people to participate. The more reputation one gains, the more power one has to moderate the site. And one gains reputation primarily by receiving upvotes. Voting is the most important form of moderation here, so restricting it to the small proportion of visitors who are logged in and have 15 reputation may seem like a bad call. However, there's a good reason for it: it helps to prevent frivolous or abusive voting intended to increase or decrease user reputation rather than indicate the value of the post. Restriction of voting and other moderation powers hopes to make the site as useful as possible to visitors.
Actually, all votes are counted, it's just that only votes by accounts with at least 15 reputation affect the score of the post and the reputation of its author. The other votes are stored in the database and can be accessed using the Stack Exchange Data Explorer. I find this feedback very useful and I wish it were more easily visible, so I like this query.
If you really want to bump the score of a post up, the easiest way to gain reputation is by suggesting edits. Many posts need basic formatting fixes and it won't take you long to rack up those few points.
The short answer is: Because that's how the site works.
Here is a list of privileges you can earn, and upvotes are earned at 15 reputation.
The idea here is that you are required to invest some effort into the site to get certain privileges - for instance by asking, answering or suggesting edits.
For general questions and answers about how the entire Stack Exchange model works, Meta Stack Exchange would be the place to visit.