Translation:My family loves me.
的 after a noun indicates that the next noun is possessed by the first. It is optional only when the possessor is particularly close relationship-wise to the subject, like family (mother, father, siblings, spouse,) and sometimes extended to a boyfriend or girlfriend. I think the sentences sounds better and more balanced if the 的 is included because it makes two characters per phrase (我的) (家人) but someone would understand you if you omitted it. This is just what I learned from my Taiwanese professor.
很 is a modifier to an adjective that can mean "very." It does not carry much meaning often in casual conversation. If something is just good/you are doing well, you would always use 很好 and never only 很 or 好 by themselves. It is a fixed expression. Although it does not carry direct meaning when translated to English, it is proper grammar. 很 gains more meaning if you use it to describe something very bad, or something extremely good. Another fixed expression is "I am busy." 我很忙。you are maybe not actually "very" busy but it must be included for grammar purposes.
As feefeehan said, it has never only meant romantic love, that's just what a lot of people seem to link the word "love" to. I remember doing so myself back in the teenage years, as that's mostly what I had in mind.
But it also means a whole lot of other things, like "I love sports", "This feeling is lovely", "Chinese is lovely to learn", "I love my family" or "I love the taste of noodles".
I think "my family loves me" is grammatically correct because it is a collective noun, though "my family love me" is in widespread use and few would bat an eyelid. In fact similar uses are made by Duolingo itself elsewhere. So flag it as "my answer should be accepted". To illustrate: we say "our families love us" because more than one family; so logic implies that "one family loves..." My opinion only. If you're an expert, feel free to correct me.
In English "loves" comes after singular things and "family" gets lumped into that category as would "audience." A collective group would be a singular thing in this case but two or more of the things would be plural. -My dad loves me -My sister loves me -My family loves me -The audience loves me -My husband's family loves me
"Love" comes after plural things. Not a collective group, but multiples of the same thing: -My dads love me -My sisters love me -Both families love me -All the audiences love me -My husband's parents love me
The "s" either needs to be on the subject or the love but not both and not neither.
It depends on British versus American English. In America (USA, I should say), groups are counted as singular, while in the UK, groups are counted as plural. So for example in the US, you would say "the FBI is watching me" while in the UK you would say "the FBI are watching me"
Then I must have a lot of American blood in me! (I think I'm probably a fussy exception to the UK 'rule'.) Agreed, BBC News always(?) has, "The Government are doing this, that or the other." Anyway, WHY is the FBI watching you; and why are they watching me? And I just noticed my inconsistency: I didn't ask, "Why is it watching me?" (Which one do Americans use?)