Once you have a picture attached, you can click on the paper clip icon and then click on the attachment you want, and it inserts it in the body. If I am remembering right, been a while since I did that.
Tapatalk basically just uploads your photo to their server and inserts it as a link, but it's all seamless.
computer, you need to use a photo hosting site like imgur, smugmug, etc. Then they will provide you with an embedding code for different ways to embed. you will want to use the one for message boards. paste that link anywhere in your reply or post and thats where the image shows up.
Just be mindful of the size (resolution) of photo you link. Too big and it messes with the whole page. I like to keep them around 1200 x 800. Seems to fit most screens without having to rescale the text to fit
The photo you linked to is 1500 x1124 and it is a little too big so it sticks out passed the regular thread boundaries