anbauen
to grow
To plant and care for plants, especially crops such as vegetables or grain, in a field or garden.
Dieses Jahr wollen wir im Garten Tomaten und Gurken anbauen.
This year we want to grow tomatoes and cucumbers in the garden.

to grow / to add on
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to grow
To plant and care for plants, especially crops such as vegetables or grain, in a field or garden.
Dieses Jahr wollen wir im Garten Tomaten und Gurken anbauen.
This year we want to grow tomatoes and cucumbers in the garden.

'Anbauen' is a separable verb. The prefix 'an-' goes at the end in a main clause. In subordinate clauses and in the infinitive with 'zu', the verb stays together.
Mein Nachbar baut sein eigenes Gemüse an.
My neighbor grows his own vegetables.
You can recognize the meaning of 'anbauen' by the object. If it refers to plants (vegetables, herbs), it is about agriculture. If it refers to parts of buildings (garage, balcony), it is about construction.
Wir bauen dieses Jahr Kartoffeln an.
We are growing potatoes this year.
In an agricultural context, 'anbauen' is often used with crops. Typical combinations are: 'Gemüse anbauen' (to grow vegetables), 'Getreide anbauen' (to grow grain), 'Kräuter anbauen' (to grow herbs), or 'Wein anbauen' (to grow grapes / cultivate wine).
In dieser Region wird seit Jahrhunderten Wein angebaut.
Wine has been grown in this region for centuries.
In a construction context, 'anbauen' is combined with parts of buildings. Common phrases are: 'eine Garage anbauen' (to add on a garage), 'einen Wintergarten anbauen' (to add on a conservatory), or 'einen Erker anbauen' (to add on a bay window).
Sie haben an ihr Haus einen Wintergarten angebaut.
They added a conservatory onto their house.
to expand
'erweitern' and 'anbauen' can both mean that a building becomes larger. 'erweitern' is more general: you can expand a house, a room, or even a range of services. 'anbauen' fits especially when a new part of a building is built directly onto an existing house. Learners may confuse the words because both can be used with houses. As a practical rule: when it is about an additional part on a building, 'anbauen' sounds very natural. When you are saying generally that something is becoming larger, 'erweitern' fits better.

Meine Eltern haben an ihr Haus ein kleines Zimmer angebaut.
My parents added a small room onto their house.
Die Schule erweitert im Sommer ihre Bibliothek.
The school is expanding its library in the summer.