Strictly speaking, it is not supposed to. It is a method of social and financial organization that allies itself to the individual motivations which help determine human action. It does do two very important things, however.
In the first place it creates the wealth that allows for welfare provision. Under a free economy people in society can become rich enough to afford higher levels of care for those in need. They may do this through charitable organizations, or they may do it collectively through government. Non-capitalist economies tend to achieve poorer results, and cannot usually afford so high a level of provision.
Secondly, the free economy itself reduces the need for welfare by a variety of market provisions. Recognizing the demand, people respond with insurance policies, health plans and pension schemes, all of which reduce the need for welfare. By encouraging people to make their own provision wherever possible, the free economy reduces the need for state welfare. Self-provided welfare can usually be tailored more closely to each individual's circumstances, whereas collective provision is often provided on standardized levels based on what are perceived as average needs.
Paradoxically, it can be the state welfare system which makes people dependent upon it. It takes the funds to support its provision which people might otherwise have used to pay for their own. In other words, high taxes make people no longer capable of providing for their own welfare. Two-thirds of those receiving benefits in Britain actually pay more in direct and indirect taxation than they receive back from the state.
Furthermore, state services crowd out private choices for many people. Private education, healthcare and pensions compete with state options which are 'free' at the point of consumption. Private alternatives charge fees, but compete with state services which do not (because they have been paid for through taxation). This severely restricts their availability and accessibility for most people.