Why do some articles on the Web have "Continue Reading" buttons?

Because there is more to read. Either you are interested in the article or not. No one other than you knows.

To make you purchase a subscription to their website.

If you’re not interested, they don’t send out the entire text. On a widely used server, this probably helps it not get overwhelmed.

Generally I’ve seen that when they want you to pay for a subscription to see the rest as @RedDeerGuy1 says.

Sometimes it throws me into ads

Generally, it’s because the news source has the opportunity to deliver even more advertising on pages 2 and 3 and so on.

I get those all of the time and it’s never about buying subscriptions. It’s usually so they can get more headlines and a couple of paragraphs in, then comes the next headline/paragraph. I don’t go to sites that require a subscription like the New York whatever-it-is or the Washington whats-its-name. I almost always read news from MSN.com.

It’s to generate as much ad income as they can. Advertisers pay per ad load, and with some stipulations about the circumstances. It’s designed to play that game as efficiently as they figure they can get away with, while driving away only what they calculate to be the most efficient number of viewers, etc.

Click bait

No it’s not @NoMore.

@Dutchess_III it can be – you can’t make an across-the-board statement like that accurately.

It can be. But is usually not
.

I think it’s so they can count how many people are reading, and use that information to sell more ads.