Le fiancé de Marie est nerveux, mais leurs fiançailles rendent toute la famille heureuse.

Fiancé is the masculine form, used for an engaged man.
Fiancée is the feminine form, used for an engaged woman.

So Le fiancé de Marie means Marie's fiancé. If the sentence were talking about Marie as the engaged person, you might see Marie est fiancée.

They are related words, but they do different jobs:

• un fiancé = an engaged man
• une fiancée = an engaged woman
• les fiançailles = the engagement

So in this sentence:

• Le fiancé de Marie = the person
• leurs fiançailles = the engagement itself

Because nerveux agrees with le fiancé, which is masculine singular.

French adjectives usually agree in gender and number with the noun they describe:

• un homme nerveux
• une femme nerveuse

Here, the person being described is Marie's male fiancé, so nerveux is correct.

French usually expresses this kind of relationship with de:

• le fiancé de Marie = Marie's fiancé
• literally: the fiancé of Marie

This is a very normal French structure. You could also say son fiancé if the context already makes it clear that son means her.

There are two important points here:

1. Fiançailles is plural, so the possessive must also be plural: leurs
2. The engagement belongs to both people, not just one person

In French, possessive adjectives agree with the thing possessed, not with the number of owners.

So:

• leur mariage = their wedding/marriage
• leurs fiançailles = their engagement

Ses fiançailles would mean his/her engagement, referring to just one person as the possessor.

Because French normally uses the plural form: les fiançailles.

This is just how the language works. Even though English uses the singular word engagement, French usually says fiançailles in the plural.

So leurs fiançailles is best translated as their engagement, not their engagements.

Because the subject is leurs fiançailles, and fiançailles is plural.

The verb rendre here means to make in the pattern:

rendre quelqu'un heureux = to make someone happy

Since the subject is plural, French uses the third-person plural form:

• les fiançailles rendent
• not les fiançailles rend

Because heureuse describes toute la famille, not fiançailles.

In the structure rendre + object + adjective, the adjective agrees with the object that is being made something:

• rendre Marie heureuse = make Marie happy
• rendre les parents heureux = make the parents happy

Here, the sentence says:

• leurs fiançailles rendent toute la famille heureuse

So the people being made happy are toute la famille.
La famille is a feminine singular noun, so the adjective must be heureuse.

Toute la famille means the whole family or the entire family.

Toute is the feminine singular form of tout, and it agrees with la famille, which is feminine singular:

• tout le monde = everyone
• tout le jour = the whole day
• toute la famille = the whole family
• toutes les personnes = all the people

So toute is used because famille is singular and feminine.