What is the purpose of this cloth in hotel?

It's called a “bed runner“.

The main purpose is to make the bed look nicer and more stylish. Where I live, it's become trendy (again?) to put something similar on tables too (a “table runner”).

Earlier, it would be common in some countries to get a similar effect by covering the bed with a large blanket, folded at 2/3rd of the length to reveal another pattern.

I have no evidence of it, but I always thought it was there to protect the bed from dirty items like your luggage or your shoes. Your luggage might be dirty because it touched the floor, so if you want to open it you put it in this piece of cloth and only that will get dirty and not the sheets where you'll sleep later. Same thing to your shoes, suppose you want to lay down for a few minutes and you're already dressed. You can lay your feet on that piece of cloth and not in the sheets.

The purpose of the bed runner is very simple: to protect the sheets. So that when you lie on the bed with your shoes on, the shoes would not dirty the sheets.

It also looks pretty, but I think that's a secondary consideration. It's not solely aesthetic.

My personal epiphany towards the use of bed runner - is for people to jump onto the bed without removing their shoes. Your legs (with shoes on) should then be placed within the area of the bed runner so as not the dirty your bed sheet.

All very good and interesting and partially correct answers.

The cloth in question originated in the early days of the medieval time period where they warmed the beds with hot rocks and coal type bed warmers stuffed in between the mattresses diligently monitored by Noble's personal help. The cloth, although I forget specific original name, was at the time actually a form of tapestry of the family crest meant for fashion of course and also was truly meant to hold the heat in the sheets as much as possible, that's the original intent.

But, over time it has taken on the name of "bed runner" and the shape of fashion. Many different and interesting stories have been focused around this item of the high-end hotel industry. Many hotels, especially in the European countries still practice this fashion statement out of respect for its true intention (without the rocks and coals, of course) and some simply like the design aspect and some wrap the cloth because it starts interesting and memorable conversations at the breakfast buffet and also because memorable stories bring you back to visit again, so it's become a fashionable piece of advertising and has taken on many true and correct reasons to add this odd item to the laundry staffs checklist.

I'm no expert of the period but I am a traveler of the world touching down at over 1350 hotel stays in my travels of business and pleasure and have been part of this conversation at many a buffet table. I was told this story at one point and actually found it to be very true, as my curiosity led me on an interesting voyage of truth of the tapestry, whose name I can't remember.