Spider Solitaire is one of the most popular solitaire card games on the planet. It can be difficult for someone who hasn't played it to figure out how to play just looking at it. You're trying to build down columns from King to Ace. When you build on the columns suit doesn't matter, so just ignore the suits of the cards. The only cards in a column you can move are the top card and any fully packed series of cards. I've dimmed all the cards you can't move so that should be easy to figure out. A fully paced series is any series of cards that is in order, as long as there isn't a card above it that is still in the series.
When you get a column of cards that runs from King to Ace, it is removed from play. When you get stuck you can deal another card to every column in the game by clicking the deck at the bottom of the game screen. When you have removed all the cards from the game you win.