sich fügen
to submit
To submit to a situation, fate, or another person's will and no longer offer resistance.
Er musste sich seinem Schicksal fügen und die Niederlage akzeptieren.
He had to submit to his fate and accept the defeat.

to submit / join
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to submit
To submit to a situation, fate, or another person's will and no longer offer resistance.
Er musste sich seinem Schicksal fügen und die Niederlage akzeptieren.
He had to submit to his fate and accept the defeat.

The verb 'fügen' (to join/submit) is related to the noun 'die Fuge' (the gap or joint between two parts). The basic idea is always that something is made to fit or is connected, whether materially or figuratively.
Der Handwerker fügte die Holzteile sorgfältig zusammen.
The craftsman carefully joined the wooden parts together.
The different meanings are easy to recognize from the sentence structure: The reflexive form 'sich fügen' means 'to submit'. The impersonal form 'es fügte sich, dass...' means 'it so happened that...'. The simple form 'fügen' means 'to put parts together'.
Er fügte sich in sein Schicksal.
He submitted to his fate.
In the meaning 'sich unterordnen' (to subordinate oneself/to submit), 'sich fügen' is used reflexively. The thing one submits to (e.g., a fate, an order) is in the dative.
Die Mitarbeiter mussten sich der Entscheidung des Chefs fügen.
The employees had to submit to the boss's decision.
The formulation 'es fügte sich, dass...' (it so happened that...) is a fixed expression. It introduces a subordinate clause with 'dass' and describes a chance or fateful event. This expression belongs to an elevated style of language.
Es fügte sich, dass genau an diesem Tag ein Experte für das Problem anwesend war.
It so happened that an expert on the problem was present on that very day.
to connect / to bandage
'verbinden' and 'fügen' can both mean that parts come together. 'verbinden' is more general and fits cables, routes, devices, or parts that are technically or physically connected. 'fügen' places more emphasis on individual parts being fitted together properly, for example in a puzzle, pieces of wood, or components. Learners may confuse the words because both have to do with putting things together. As a simple guide: when a connection is created, people often say 'verbinden'; when parts go exactly into their place, 'fügen' is more likely to fit.

Lisa fügt die letzten Teile des Puzzles zusammen.
Lisa fits the last pieces of the puzzle together.
Der Techniker verbindet den Drucker mit dem Computer.
The technician connects the printer to the computer.