How often is really hard to say. Southern law in the later antebellum often requires the state legislature to sign off and the manumitter to post bonds and grant assurances of the freedperson's good behavior or removal from the state. That creates a paper trail, but the relative visibility of such a trail in the archives can elide options like taking slaves to a free jurisdiction and just letting them go or helping them settle there. One of Thomas Jefferson's younger relations, Edward Coles, did the latter with some slaves he inherited. (Against TJ's advice, incidentally.)

But with the possible exception of late slavery in Delaware and some other marginal spaces, self-purchase isn't available on such a scale that it's a threat to the system. As such, it doesn't generate the reflexive horror that things like antislavery politics do. Self-purchase and other forms of manumission and emancipation are generally more common in the Upper (non-Cotton) South than the Lower South because the subsection ends up with more slaves than it really needs due to the decline in tobacco profits. Flush with the cotton boom, the Lower South doesn't have that problem. It's also probably financially easier because there are larger free black communities dating to the Revolutionary era who might help chip in.

Every case is different, but the biggest thing to remember about self-purchase is that it's entirely at the whim of the enslaver. They can change their mind at any time and can't be held accountable to past promises because the slaves don't have any legal recourse, except maybe in Louisiana as a holdover from French practice. Those who give their word and keep it have multiple reasons.

The price to buy a slave is set by the enslaver, who can dictate terms that allow an effective profit off the transaction. If the cost is high enough it may also take years or decades to meet, which are all money in the bank for the enslaver too. A middle-aged slave is worth less for most uses than one in his or her prime, so letting the slave buy out can be a way to convert a depreciating asset into something (or someone) more valuable. If the slave is actually elderly, then that goes double. For enslaved women, passing their childbearing years decreases their value since much of it is bound up in their "increase", or ability to produce new slaves for free through normal reproduction.

Offering self-purchase can also be an inducement to better performance by the slave, a carrot to go with the horrific forest of sticks that build up the house of bondage. By making the promise contingent on good behavior (not running away, working hard, etc) the enslaver might get more from the slave until the day of freedom. Securing against slaves stealing themselves off to freedom is a significant hedge particularly near to free states, since a slave who gets away is just lost money. That can have knock-on effects for other slaves owned by the same person, since they might believe the option exists for them as well.

This does mean a lot of money, though. Enslaved people got that money by a variety of means. A lot of the cash would come from the semi-illicit slave economy. Custom and practicality dictated that slaves tended to grow their own food, and occasionally some cash crops, on land provided by the enslaver. They did this on their own time, often at night or when otherwise "off". The law did not recognize the property rights of enslaved people, but ordinary people on the ground didn't care that much. If a slave wants to sell you something you want and the price is right, why not buy it? Skilled slaves might already have something like their own businesses and be allowed to largely hire their own labor in exchange for hefty payments to their enslavers. The revenues from those trades and any surplus farm produce can all go into the owner's pocket at any time, but what didn't could be saved against freedom. What one doesn't have oneself from those streams might come in through friends, family, and other social ties in the area who have the same or better access. Self-purchase often had a community element.

Abolitionists buying slaves their freedom happens too, but it's not quite an ordinary part of antislavery activity. For the most part, groups pitch in cash to buy slaves who have already escaped slavery but are at risk of or have been recaptured. I've never heard of a case where they gathered funds to hire a person and to go down to Virginia and buy at auction or anything like that. When someone goes south to buy someone for freedom, it's usually a specific person who they either know or the relative of one they knew. Even funding underground railroad activities came under fire from some white antislavery quarters for not doing enough to bring down slavery as a system, though for the most part raising money to help fugitives get to safety seems to have been an accepted (if always inadequately funded) part of antislavery activism.

Sources

Many Thousands Gone by Ira Berlin

Gateway to Freedom by Eric Foner

Road to Disunion, Volume One by William Freehling