I can see how a new product can value more than the sum of its parts, but I don't understand how this can increase the wealth of a society.
The wealth of society, in material sense, consists of what it can produce. Production of more products or services is the wealth. Economic growth is by definition a continuous increase in production. In fact the way how we most commonly measure economic growth is to just measure output per person by measuring GDP per capita.
Wealth is not money, money is just a measurement stick for value, storage of that value, and tool to simplify exchanges, but it’s not wealth in itself. However, because it is so ubiquitous many non economists equate it with wealth itself but that is just ‘money illusion’. If you would have billion \$ but could not buy even a loaf of bread with it you would be poor compared to a person with 0 dollars who has access to any goods or services they wish.
let's say a person can spend only what she earn as wage, so when a new product come out, this product is competing with all the pre existing products, someone should lose to someone else to win.
In competitive market economy people earn approximately what is their marginal product. When you go to work you either create goods or services for someone directly or indirectly and in competitive market you will get for that money equal to the worth of your marginal product. If you can produce more with the same or less inputs you will also earn more money.
The reason why for example Solow-Swan model shows that economic growth depends on technology is that technology makes people continuously more productive year after year and allows them to produce more. This increases production is the real wealth, but besides that if people are more productive they also get pay more.
In real life not all markets are competitive so the above holds only approximately but empirical evidence shows that real wages indeed track net productivity over long term.
Besides increases in technology endogenous growth theory suggests that also savings rates and support for basic R\&D and human capital can accelerate economic growth. The mechanism there is different than in Solow model but ultimately again what delivers growth is how the above boosts the productivity and output.