Here are some examples:
* "Il pleut, ainsi je reste à la maison." (It's raining, thus I'm staying home.)
* "Elle a travaillé dur, ainsi elle a réussi son examen." (She worked hard, so she passed her exam.)
* "J'ai mal à la tête, ainsi je vais prendre une aspirine." (I have a headache, therefore I'm going to take an aspirin.)
* "Le soleil brille, ainsi je vais aller me promener." (The sun is shining, consequently I'm going for a walk.)
In essence, "ainsi" indicates a connection between the previous statement and the following one. It shows that the latter statement is a result of the former.