Why can early embryos survive freezing?
The reason why cells (and tissues) die during freezing is that ice crystals rupture the cell membranes1,2, so to me it seems the question is actually why early embryos can survive this process.  

The answer depends on the technique used, but it comes down to size.  Small tissues (or cells) have a large surface area to volume ratio.  

In techniques that use cryoprotectants (compounds that suppress ice crystal formation), the small size allows efficient replacement of water by cryoprotectant prior to freezing (and the reverse after thawing) so that the cells don't have time to die1,2.

In flash freezing (vitrification), the small size allows the entire tissue to be frozen instantaneously, which also suppresses ice crystal formation1,2.

This Scientific American article is very relevant as well. 



References:
 1: Konc, J., Kanyó, K., Kriston, R., Somoskői, B., &amp; Cseh, S. (2014). Cryopreservation of embryos and oocytes in human assisted reproduction. BioMed research international, 2014.

2: Loutradi, K. E., Kolibianakis, E. M., Venetis, C. A., Papanikolaou, E. G., Pados, G., Bontis, I., &amp; Tarlatzis, B. C. (2008). Cryopreservation of human embryos by vitrification or slow freezing: a systematic review and meta-analysis. Fertility and sterility, 90(1), 186-193.