After scrolling to select the axe, the yellow box will highlight it in the toolbar. That is all that is needed to select and be holding the axe, even though you can't see it. "To break blocks" refers to breaking blocks in the game, and you need to hold left click while hovering over the collision area of the block(the part of the block that the player and other creatures can't travel though) Blocks include trees as well as any blocks you craft from the list of recipes provided on the home page of the website, and includes beds, wood blocks, and troughs, etc. Blocks are any type of item that can be placed down when selected in the toolbar. Placing blocks requires the block to be selected in the toolbar, and then a placement preview for the block will appear where your cursor is when it is near the player. Right clicking will place the block in the location that the preview was in previously.
When a block is destroyed it will simply disappear from the world and appear in your inventory. The exact opposite happens when placing a block. To summarize, scroll so that the axe is highlighted in the toolbar. then hold down left click with the mouse cursor over the base of a tree nearby the player, so the player can reach the tree (the base is where the collision area is). The tree should break after a second or so and will appear in your inventory.
I hope this covers everything you need, but let me know if you need more instructions.