Populating list of dates from another sheet with Start Dates and End Dates

Hi, welcome to the forum

To get the 1st name is easy...
=IFERROR(INDEX(Sheet1!$A$2:$A$4,MATCH(Sheet2!A2,Sheet1!$B$2:$B$4)),"")
To add additional names will get tricky. What would the max amount of extra names be?

I have taken a slightly different approach on this, using Conditional Formatting to highlight the date range in each row - it shows up the "overlap" really well, not sure is this is something you can work with?

Try array-entering this in C2 and filling down.

Formula:

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If you aren’t familiar with array-entered formulas array enter means the formula must be committed from edit mode by simultaneously pressing and holding down Ctrl and Shift while hitting Enter.

Row\Col	A	B	C	D
1	Date	Project(s)
2	11/1/2015			{=IFERROR(INDEX(Sheet1!$A$2:$A$4,SMALL(IF((A2>=Sheet1!$B$2:$B$4)*(A2<=Sheet1!$C$2:$C$4),ROW($1:$3)),COUNTIF($A$2:A2,A2))),"")}
3	11/2/2015	Project A	Project A
4	11/3/2015	Project A	Project A
5	11/4/2015	Project A	Project A
6	11/5/2015	Project A	Project A
7	11/5/2015	Project B	Project B
8	11/6/2015	Project B	Project B
9	11/7/2015	Project B	Project B
10	11/8/2015	Project C	Project C