Why the tape on framed paintings and prints?

And why is it going brown and brittle?

Should I fix it?

And can I do it at home?

They arejust prints under glass with a matt thingy.  But amazing prints that are close to my memory of the actual colours of the paintings.  None of this distorted colour like in art books.  I love them and shall adorn my wall with them.  I wouldn't mind if they lasted another 50 years.

Also, am I supposed to write my part of the history on the back?

That is interesting. When we got some of my wife's art professionally framed, it did not have tape, but had brown paper that was glued flat across the back from frame edge to frame edge. I assume the tape would be for helping to hold the matte in place without using glue?

The brown paper system seems to hold up better.

Can that be done at home?  And would normal craft paper work or would it need fancy paper?

I want these to last the rest of my life.

I don't KNOW why the tape (or brown paper), but I'm going to guess.
What if it is there to moderate humidity?

Way back, a life time ago, I worked in a framing shop, cutting mats, the works.
What I remember, the final seal for the back for prints and originals alike was brown paper, cut to approximate fit, glued to the edge of the routed area, seems like elmers glue? barely dampen the whole paper with a sponge (to cause it to stretch nice and tight when dry) trim the excess with a razor blade when dry,( or mostly dry?)

I've done this since for original watercolors and prints and it makes a nice neat long lasting backing.... have not tested for fifty years yet though and the glue might be the key...and acid free paper?

For yours, where there is writing that you would want to preserve, I suppose the tape could be replaced with a narrow band of brown paper and similar method?
As others have said the paper is to seal out dust, moderate moisture and keep out insects to a certain extent.

I doubt, depending on the glue and type of paper, that it would prevent certain paper and glue eating insects.

I have been a bespoke picture framer for the last 30 years. Working on gallery grade faming.
The mount-board (coloured cardboard) is used to highlight the picture and give it depth, also to hold the painting away from the glass and stop discoloration. If professionally done, the mount will have a slightly deeper lower edge, to balance the picture when hung on the wall. The mount-board will (should) be made from high quality acid free paper, also to stop discoloration and damage to the picture.
The brown tape at the back of the frame is essential to elevating humidity and dust entering the frame, extending the life of all parts involved the framing process.
Normally this tape is a professional framing tape, that sticks to the molding (wooden bit) but not to itself, but any brown tape can be used, self-adhesive or gum tape. If using gum-tape make sure most of the moisture has been removed before applying it to the back of the frame.
As for leaving a message in the back if the picture frame, I always loved this when reframing a picture. The history and love we can share with future art lovers, just makes the art feel so much more special.

For most of my framing years we used a 3M product called ATG to attach the paper backing. It's basically double sided tape but the glue only, no additional body. Comes in rolls and we used the 1/4" size. You lay it down along the frame back edge, press it down and peel off the backing then lay down a piece of kraft paper that is a bit larger than the frame. Press the paper down and then use a razor blade to cut off the paper to the exact edges of the frame. Gives a really clean back and no waiting or added moisture.