First of all:
Note, that in German grammar there is nothing like direct or indirect objects. Open any German grammar book that is written in German language (i.e. for German native speakers). You will not find the term »direktes Objekt« or »indirektes Objekt« in any of those books. This terms doesn't exist in German grammar, so please don't learn them. They can be helpful for a beginner, but they are misleading in many cases.
Please learn what really exist, i.e. accusative object, dative object, genitive object, prepositional object (and some authors also use the term nominative object as synonym for predicative nominative).
Now for your question:
The two sentences
Dem Alter ein Erinnerungsgarten,
To the seniority a garden of memories,
Der Jugend ein Entdeckungsfeld.
To the youth a field of discoveries.
are ellipses, i.e. incomplete sentences. As you can see there is neither a subject nor a verb in them. So the reader has to add both.
But you should use ellipses only when it is clear to everybody what is left out. And here it is clear.
This is the sentence before (subject and verb are marked bold):
Für jeden bin ich eine Welt!
For everybody I am a world!
Here we have a subject (»I«) and also a verb (»am«). And both fit perfect into the two ellipses:
Dem Alter bin ich ein Erinnerungsgarten,
To the seniority I am a garden of memories,
Der Jugend bin ich ein Entdeckungsfeld.
To the youth I am a field of discoveries.
What also might confuse some learners of German is the kind of use of dative case that doesn't exist in languages like English: Dativus commodi.
This use of dative case marks the dative object as the beneficiary or victim of the action. In the first sentence (»Für jeden ...«) this role was marked with a prepositional object, but in the ellipses the author used dative objects (to be interpreted as dativus commodi) for the same semantic aim.
Also note, that what might look like a subject in the ellipses (»ein Erinnerungsgarten« and »ein Entdeckungsfeld«) are not subjects. In German you call this grammatical construction »Gleichsetzungsnominativ«. It is some kind of nominative object (some authors really use this name because it can be interpreted as an object), but more common in the term predicative nominative because it can be interpreted as a part of the predicate.