On my (cheap) QW TB-1207 Android 4.0.3, it is a bit hard to find the external SD card too (in BSPlayer's browser mode).
When using FileManager you can directly (at the top bar) select either internal or external SD card (or USB or ... etc.).
However, BS.Player (in browser mode) will show you the internal SD-card with no fuss at all, but it takes a bit more to find the external SD-card.
First you (repeatedly) tap "Browse" till you get at the top (or root) of the tree.
Then, on my device, I have to tap on "mnt" (I assume this stands for "mount") and then select "extsd" ("sdcard" is also shown under "mnt", but that's the internal one).
What path further to follow - of course - depends on the structure you may have created yourself on your SD-card.
Maybe this is similar on other tablets, the owners of which have trouble to find their external SD-card using BSPlayer's browser?