Generally, Hindus are vegetarian. They avoid mamsah i.e flesh of animals like meat, fish - because it means their suffering and death. Some avoid eggs also. Atleast it is very limited, and they don't eat it daily like western countries, and avoid it during religious functions.

What is the reason behind this ? Is compassion/ahimsa the only reason ? Is it voluntary or compulsory ?

Is there any punishment for eating meat ? Is there any reward for avoiding meat ?
What is the real meaning of the word 'mamsah' (मांसः)? Does it have anything to do with why they are vegetarian ?

Eating meat outside of rituals is considered as a sin unanimously in all scriptures.. that's why probably.

Manu Smriti 5.31. ’The consumption of meat (is befitting) for
sacrifices,’that is declared to be a rule made by the gods; but to
persist (in using it) on other (occasions) is said to be a proceeding
worthy of Rakshasas

5.33. A twice-born man who knows the law, must not eat meat except in conformity with the law; for if he has eaten it unlawfully, he will,
unable to save himself, be eaten after death by his (victims)

5.48. Meat can never be obtained without injury to living creatures, and injury to sentient beings is detrimental to (the attainment of)
heavenly bliss; let him therefore shun (the use of) meat.

Also abstaining from eating meat bestows spiritual benefits.

5.56. There is no sin in eating meat, in (drinking) spirituous liquor, and in carnal intercourse, for that is the natural way of
created beings, but abstention brings great rewards.

As for punishments, we have the following Manu Smriti verse:

5.38. As many hairs as the slain beast has, so often indeed will he who killed it without a (lawful) reason suffer a violent death in
future births.

Without lawful reason -- that is killing animals outside of rituals.

For your query about the word Mamsah see the following:

5.55. ’Me he (mam sah)’will devour in the next (world), whose flesh I eat in this (life); the wise declare this (to be) the real meaning of
the word ’flesh’(mamsah).

Manu Smriti:

'Me he (mam sah)' will devour in the next (world), whose flesh I eat
in this (life); the wise declare this (to be) the real meaning of the
word 'flesh' (mamsah).

He who permits (the slaughter of an animal), he who cuts it up, he who kills it, he > who buys or sells (meat), he who cooks it, he who serves it up, and he who eats it, (must all be considered as) the slayers (of the animal).

He who during a hundred years annually offers (Ashvamedha), and he who entirely abstains from meat, obtain the same reward for their meritorious (conduct).

Mahabharata - Anushasana Parva -Dana Dharma Parva - Chapters 114-116 are discussing this issue in very detail. This is discussion between Bhishma , Yudhisthira and Rishi Vaishampayana. These chapters is highlighting the topic of Ahimsa (non killing).

what is the real meaning of the-word-mamsah-मांसः?

The etymology of word mamsah :
The essence of word Mansa is told by Bhishma in Chapter 116 - Verse 25.

मां स भक्षयते यस्माद भक्षयिय्षे तपप्यहम |
एतन्मासस्य
मांसत्वनुबुद्ध्यस्व भारत ||25||

P 243 - Since he hath eaten me, I shall eat him in return,--even this,
O Bharata, constitutes the character as Mansa of Mansa. So the real
meaning of mansa is he creature who is going to eaten says that , i
also in future will eat the eater in same manner. And this is the
real meaning of Ma मां Sa सा.

Footnote - Mansa is flesh. This verse explains the etymology of the word, Mam (me) sa; Me he eateth, therefore, I shall eat him. The words following Me he should be supplied in order to get at the meaning.

According to Mahabhrata the meaning of Mansa is the flesh of those who are born in womb or from the womb. And there is no difference between flesh of persons own son and flesh of other creatures.

What is the reason behind this?

According to Bhishma there are four ways to follow Ahimsa Dharma. By Mana (mind) , by Speech , by Karma and not eating flesh . So all these are part of Ahimsa Dharma and not eating flesh is one of them.

चतुर्विधेयं निर्द्रिष्टा ह्यहिन्सा ब्रह्मवादिभि : |
एकैकतोSपि
विभ्रष्टा न भवत्यरिसूदन ||4||

"Bhishma said, 'Utterers of Brahma have said that there are four kinds
of compassion or abstention from injury.

i.e. even by mentally mentally committing an act of slaughter, one becomes guilty of it. So those who are wise avoid it because to follow Ahimsa Dharma.

Is there any reward for avoiding meat?

Yes there is certainly a reward for avoiding meat. It is said in mahabharata that a person should avoid meat for his entire life and by doing so he will get the highest place in swarga i.e. heaven.
But apart from that it is also meritorious to leave the flesh in Shukla paksha -Sharad Ritu and that will become dharma. Not eating meat for 4 months brings fame , long life and strength. And for one month healthy life and freedom from all sorrows.

सर्वमांसानि यो राजन् यावज्जिवं न भक्षयेत |
स्वर्गे स विपुलं स्थानं
प्राप्नुयान्नात्र संशय : ||24 ||

P 243 Hence, a person of cleansed soul should be compassionate to all
living creatures. That man, O king, who abstains from every kind of
meat from his birth, without doubt, acquires a large space in Heaven,
They who eat the flesh of animals who are desirous of life, are
themselves eaten by the animals they eat, without doubt

And as it is part of Ahimsa Dharma which is itself is part of Dharma of man , so Hindus avoid eating flesh of animals.