I don't own the same puzzle as you, but I have a Burr puzzle set that is close. I am not completely sure if the solution I describe here will work on your puzzle.
I'll number the pieces #1 to #6 in the order you showed them in your question.
Combine pieces #4 and #6 to make rectangular ring. One end of the ring opening has a notch. Insert #5 into that ring so that it enters that notch.
Place #2 on one side of the ring, perpendicular to all the other pieces.
Piece #3 goes on the other side of the ring, to pair up with #2.
(Note, in the picture my thumb covers the piece a bit because I don't actually have a piece that matches your #3.)
The last piece, #1, cannot be put in place directly. The other pieces need to be slid about a bit to allow it to be inserted.
First slide #6 lengthwise, which disengages the notch from #5.
Now slide #5 up. This then allows #4 to move lengthways in the opposite direction that #6 went.
Pieces #4 and #6 have slid far enough apart that their ring is broken. This gives enough room to insert #1 inside the ring. Lastly, slide pieces #6, #5 and #4 back, and the puzzle should be solved (if I am visualising this correctly).