من خیلی تشنه ام؛ لطفا به من آب بده.

ام is the enclitic form of هستم (I am) and it attaches to nouns/adjectives. Many texts write it separated (تشنه ام), but a very common and more standard-looking way is to attach it with a half-space: تشنه‌ام.
So these are essentially the same in meaning:

• من تشنه‌ام.
• من تشنه ام.
  (Attaching with ـم can also happen: خسته‌ام = I’m tired.)

A natural pronunciation (roughly) is:

• man kheyli teshné-am; lotfan be man âb bede.
  Notes:
• خیلی is usually kheyli (not khili).
• The stress is often on ـنه in تشنه → teshné.
• آب is a long â sound: âb.

It can often be dropped because the ending ـم already shows it’s I:

• خیلی تشنه‌ام؛ لطفاً به من آب بده. (natural) You might keep من for emphasis (like I am the one who’s thirsty).

خیلی means very/a lot and commonly comes right before the adjective:

• خیلی تشنه‌ام = I’m very thirsty. You can also use خیلی with verbs (meaning a lot), but here it’s modifying an adjective.

The first من is the subject (I).
به من means to me and marks the indirect object of دادن (to give):

• به من آب بده = Give water to me. In everyday speech, به من often becomes بهم:
• لطفاً بهم آب بده.

Persian is typically SOV (Subject–Object–Verb). The object usually comes before the verb:

• آب بده (object + verb)
  So به من آب بده is a normal Persian word order: to me + water + give.

را is an object marker, but it’s not always needed—especially in casual requests and when the meaning is clear.
Both can be correct, with slightly different feel:

• لطفاً به من آب بده. (very common, natural)
• لطفاً آب را به من بده. (more explicit/emphatic, sometimes more formal)

بده is the informal singular imperative of دادن (to give). You’d use it with friends, family, children, etc.
More polite/formal options:

• لطفاً به من آب بدهید. (formal/plural)
• میشه به من آب بدید؟ (very common polite request in speech)
• لطفاً به من آب می‌دهید؟ (polite question)

لطفاً means please and is common with requests. It can appear at the start or before the verb:

• لطفاً به من آب بده.
• به من لطفاً آب بده. (possible, but the first is more common)

It’s correct and understandable. Very natural spoken versions include:

• خیلی تشنه‌ام؛ لطفاً بهم آب بده.
• خیلی تشنه‌ام؛ میشه یه کم آب بدی؟ (Could you give me a little water?)
  Adding یه کم (a little) or یه لیوان (a glass) can sound more situational.

It separates two closely related clauses: 1) من خیلی تشنه‌ام
2) لطفاً به من آب بده
It’s not required; you could also use a period or comma:

• من خیلی تشنه‌ام. لطفاً به من آب بده.
• من خیلی تشنه‌ام، لطفاً به من آب بده.

You can say آب بده if the context makes it obvious who should receive it, but it becomes less specific:

• آب بده = Give (some) water (to me / to someone—depends on context)
  Adding به من removes ambiguity:
• به من آب بده = Give water to me.