Fritzing parts only auto-connect if some pins are declared as male. This can be checked by looking at the part's XML file (search for
<connector.*type=) or by opening the part in the 'new parts editor' (i.e. right-click->edit).
Note that it's ok for a part to have heterogeneous connector types. For example, male ones on the long sides and female ones on the short side. Which is what's used commonly with these parts, in reality. This is how for example the M1 Blue Pill or even the Arduino Pro Mini Rev13 (5V) Fritzing parts are defined, which thus auto-connect fine, as expected.
One can only speculate why the Arduino Pro Mini Rev14 (8 MHz/3.3 V and 16 MHz/5 V) parts were defined differently. Might be a bug or a deliberate design choice. However, for being a deliberate choice it would be only consequential to also draw female up-facing headers on the underlying SVG (cf. the Arduino UNO part, where this is done).
To work around this, one can edit the part, change all the connector types on the long sides from female to male and save it as new part - say - using the 'new parts editor', directly inside Fritzing. After that change the new part shows up under the 'MINE' parts category.
After that change the part auto-connects to the bread-board. However, the alignment is still broken, thus, for auto-connection one has to disable View->'Align to Grid'.
So to also fix the alignment some more sophisticated part editing is necessary.