Sewing a button on was a near fact of life that was taught to kids as a matter of course where and when I grew up. That bit of thread wrapped around is a way of strengthening the attachment and is just the same bit of thread used to sew through the holes and cloth wound around before securing.
Check how the other buttons are attached and follow the pattern accordingly. Do not sew buttons on too tightly, they need a bit of space between the rear of the button and fabric otherwise the button hole won't sit nicely, in fact if sewn too tight you won't get the button done up properly at all.
For larger external jacket buttons an old tailor's trick that I was taught is to insert a used wooden match behind the back of the button and fabric you are sewing. Sew the button on as per the original pattern but with the match in place behind it, do not sew too tight. Once you have the right amount of stiches bring the needle and thread out between the back of the button and fabric beside the match. Carefully withdraw the match - the button will be loose but that's the aim. Carefully pull the button tight away from the fabric, pull the needle and thread out from behind the button and wrap enough thread around the threads to draw them together and form a neat shank. More difficult to explain than do but the shank and gap allows the button hole to sit and close neetly behind the button - this cannot occur if the button is sewn tight against the cloth.
For handsewn jobs like this I will draw the thread through a block of beeswax prior to sewing. This helps prevent the thread twisting and knotting when sewing and makes it run easier through the fabric. A thimble will make pushing the needle through layers of fabric easier on the middle finger.
This video will help but note they hold the match on the face of the button but I tend to hold it to the rear. Personal choice.