A Klein bottle is a 2-dimensional manifold. It can't be embedded in R3, but it can be embedded in R4. This is not what people normally mean by "4d", though. It is like a Moebius strip in that they are both 2-dimensional non-orientable manifolds, though of course a Moebius strip can be embedded in R3. The Moebius strip has a boundary, which sometimes excludes it from a strict definition of "manifold".
You can construct a Klein bottle using a rectangle with two edges with the same orientation and two edges with opposite orientation (K^2 is a good notation for this). Naturally identifying the same orientation edges gets you a cylinder. When you identify the remaining two edges in R^3 you have the classic Klein bottle we're familiar with. But there is no embedding of K^2 into R^3, but there is for R^4. This is intuitive because the map from K^2 to R^3 is not one-to-one because two circles of K^2 have the same imagine circle in R^3.
You can also look at it from the Möbius strip perspective, the Möbius strip being a twisted product (in contrast to the annulus being just the product) of S^1 x R^1, in which case some of the work is already done for you.
The surface itself is the Klein bottle, so I would expect its boundary to be some 1-dimensional object, in the same way the boundary of a disk is the circle surrounding it.
You get a Klein bottle from Möbius strip by identifying the boundary 1-cell with itself (plus a twist, whatever). It is not a "3d volume"; there is no 3-cell involved.
From wiki: "Whereas a Möbius strip is a surface with boundary, a Klein bottle has no boundary (for comparison, a sphere is an orientable surface with no boundary)." (https://en.wikipedia.org/wiki/Klein_bottle)
One standard way to construct both a Möbius strip and a Klein bottle (and other classic manifolds) is to take a square, and glue some edges together.
For a Möbius strip, you glue together the left and right (say) edges, such that the upper part of one connects to the bottom part of the other (that is, put a twist in the square before you glue).
For a Klein bottle, you additionally glue together the top and bottom edges, but don't twist them. This is what it means to say a Klein bottle is "like a Möbius strip" or "two strips glued together" or similar things.
The "4d" comes in because if you want to do this with a physical object, you need four spatial dimensions unless you're ok with it passing thru itself.
