Actually, the OP was probably viewing the spider’s interest in reverse.
I don’t know who he is and I really don’t want to meet him! Thank God I live in England where spiders may be creepy but do not have eyes as big as saucers. 
Not only do jumping spiders have good eyesight, they can also hear you.
Actually the spider I was talking about was seen in England - Zebra Spiders are quite common in England, but they’re small (around about the size of money spiders) and quite shy so you don’t see them that often.
You know, looking at this chart, it really jumps out at you how little variety there is in the eyes of vertebrates.
I had a black and white jumping spider in my window and I swear we had a bond of some sort. I used to feed him bugs and he got to where he would come out of hiding and climb on my hand. I was blown away by his intelligence and awareness. He had really grown huge for a jumping spider and I am pretty sure my ex killed it when she took over my guest house. I was furious.
That was how I got over my fear of spiders - getting to know individuals and watch their lives and appreciate their intelligence. The fear turned to fascination and now to obsession. Love them!
Ron Hoy, professor of neurobiology and behavior at Cornell University led a study that measured the brain activity of these little guys when they were shown photos of things they should recognize. They recognize things.
Having near-human eyesight it really shouldn’t be a surprise that they can tell when they are being looked at by a giant.
Since everything they do is governed by a brain the size of a poppy seed they probably aren’t doing any heavy thinking about it, but since they can see so well with such a tiny brain they’ve got some kind of something going on whether you want to call it awareness or not.
The OP’s description of their behavior is only a little bit more enthusiastic than one of the leading researchers of the subject.
Jumping spiders are on one end of the spider continuum and those horrible giant orb weavers (Areanus) are at the other. Not the Nephila orb weavers, I like those. They don’t have repulsive fat abdomens like the Areanus kind.
I Hope those things don’t live near airports… they could seriously divert a 747 on that last furlong.
“Orb weaver” is a lot larger category of spiders than most people realize. You know what a spider web looks like, right? A bunch of radial rays, with a tight spiral wound around them? That’s an orb web, and any kind of spider that makes a web like that is an orb weaver.
Pretty much every animal I can think of has a nose and mouth that faces in the same direction as their eyes. Thus, some creatures sense when they are being looked at (stalked) because they are actually sensing the carbon dioxide being exhaled. I don’t know if that pertains to this particular creature, however.
It may simply be the 360 vision afforded by their eight eyes. It allows them to keep an eye on you … and you … and you over there, etc. LOL
Not exactly. Most orb-weavers belong to the Araneidae, but a few other families also make orb-webs, including the Tetragnathidae and Uloboridae. The latter don’t use sticky silk in their webs, but instead silk coated with fuzzy fibers.
Orb-webs are the stereotypical spider webs, but tangled webs or cobwebs are more common. Spiders also make sheet webs, tubular webs, and funnel webs.
Jumping spiders don’t make webs for catching prey at all, or for securing it. They do use silk for tether lines and for making shelters, however.
The uloborids get even more confusing. Not only do they make an orb web with fuzzy (cribellate) silk, they don’t use venom. Really mess up classification. Our locals look like tiny scraps of junk in lots of webs together all over our back veranda. They attach to the chairs whenever they are left unattended. Messy little darlings.