How did slaves buy their own freedom? What stopped slave owners from simply confiscating any saved up money?

This is a pretty complicated question to answer generally as slavery varied quite a bit from urban to rural, Upper South to Lower South, etc. (I presume you're asking about American slavery). There are a few key things to remember when considering this question:

The slave experience and the quest for liberty heavily depended on movement; the ability to travel from the plantations into towns/cities; the use of movement as a form of resistance (by permanently or temporarily escaping the slaveholder); and finally, forced movement caused huge disruptions in familial life for enslaved people, as mothers and fathers were ripped from their children, who were often left with kin or non-family community members to be cared for, disrupting their path to freedom.
Community, both free and enslaved, were critical for the vast majority of enslaved people to get their freedom. Free black communities (both near and far from slavery, including those in the North) raised money to purchase slaves, provided safe havens for slaves self-emancipating by running away, and helped newly settled former slaves support themselves and find their families.
There are temporal and regional dimensions to consider. Slaves in northern New England, for example, utilized the patois of freedom during the American Revolution-era in order to hire lawyers and successfully sue for their freedom, arguing that slavery was incompatible with the new constitution, resulting in the abolition of slavery in Massachusetts, for example, in 1784. Charleston had one of the largest and most successful free black communities in the country, many of whom were skilled craftspeople and who were able to utilize their skills to help purchase the freedom of both friends and strangers. It's also important to remember that free blacks were being ever-restricted in the South, especially after the Nat Turner Rebellion in 1831, which caused states like Virginia to pass a series of laws restricting free black communities, as well as seriously restricting their behavior and ability to gather.
Slaveholding itself varied greatly, as did slave holders and traders. The nature of slavery was such that it permeated every single level of society in the South. Ira Berlin has argued that there were regional distinctions in America, an argument of a "slave society" vs. a "society with slaves." (See Many Thousands Gone 1998). In slave societies, planters, traders, and other pro-slavery people seized all political power, and slavery itself "stood at the center of economic production." In societies with slaves, such as Massachusetts pre-1784, slavery played a much smaller role in political decision making (though I would argue that you cannot divest slavery from the merchant maritime economy of the region, even well into the 19th century). Some slaveholders allowed slaves to purchase themselves without issue; others would never allow them freedom. This is why there were still millions of people in bondage in America at the outbreak of war in 1861. Freedom came at a heavy price for most.

Slavery and slave owners varied greatly. For many enslaved people on plantations, there was little to no opportunity to make money. Slaves worked long hours, and what little spare time they had was often dedicated to gardening for themselves and their families, as the vast majority of enslaved people on plantations suffered from some form of malnourishment due to insufficient food being supplied to them by planters. To offset this, many enslaved people were allowed to have kitchen gardens. Some planters would allow them to travel to towns to sell their surplus crops. Other skilled craftspeople and tradespeople were "hired out," meaning that the slaveholder would allow another person to essentially lease the slave to do a task. Sometimes this meant traveling to cities from the plantation. Often this movement, and working under circumstances that were different than chattel labor, meant that they could hire themselves out to do tasks and raise money. These people worked as carpenters, seamstresses, sailors, blacksmiths, and even entertainers, among many other things.

Of course, none of this means that slaveholders were willing to allow slaves to purchase their freedom. As slavery became more contested in the 1830s and into the Civil War, free blacks were routinely attacked politically, socially, and often physically, even in the North. This was especially true after the outbreak of war in 1861, as many whites blamed all African Americans for the devastating war that was killing scores and tearing the country apart (rather than the slaveholders/slavery, the true cause of the war).

Some sources:

Berlin's Many Thousands Gone; Anthony Kaye Joining Places: Slave Neighborhoods in the Old South; Amrita Chakrabarti Myers Forging Freedom: Black Women & the Pursuit of Liberty in Antebellum Charleston; Douglas Egerton, Death or Liberty: African Americans and Revolutionary America