Eye contact is a start.

Put the focus on their goals.

By listening to their needs.

If someone is a client, I've already been "engaged" with them for some period of time.

Questions, tell me more, then more questions.

Regular communication, client parties and one on one coffee get together, pop-bys, etc.

If they are difficult, in a Sumo Ring. 

I'm not that type of guy.

LISTENING and asking questions. 

Listen, talk to them !

human to human engagement in real estate. I only have patience for 30 seconds to hear their wishes. I ask two questions and decide to dis-engage or not.

By mastering the art of listening! It was not easy. It was a long road and the journey continues.

Find out what their needs are in real estate and deliver the goods

"Hi, I'm Jennifer"

I start by asking questions and then listening to their answers.

Mike Albert By introducing myself and developing rapport.

Of course. We search or a commonality to establish a good rapport.

Face to face contacts are the best. Discuss their needs.

Get out of cyberspace and get face-to-face.  

Nina Hollander nailed this one!

Make their needs & concerns your priority. Be a problem solver for them.

Client. That means we've meet, have reached abusiness understanding (contract) so this is, as your question states, an deepening of their engagement with the outcome of them giving their blood, sweat, tears and undying loyalty.

I give them tools they can use to assist in the real estate goals.

An example or two:

CIS Score (buyer, seller, citizen)

REIT spreadsheet (investor)

Joist app (flipper)

BEP - (buyer)

Chain them to your car till they buy!!!  I have a form for them to  fill out about their needs

Not sure I understand the question. In what context. Online? Over the phone? Who called who? While showing? At a party? -Kasey

Many different ways- email to catch up on their lives, phone call or make a lunch date.

I have two ears and one mouth, and I use them with the same proportion.

Listening and communication.

"Hi!"  Always a good start.  Also the old classic:
"How much does a polar bear weigh?  Enough to break the ice!  Hi, I'm Ryan"

I like the advice: be warm, not pushy. Grow relationship, nurture your client. Be consistent with follow up before, during and after. Use systems to track your results. 

Find out what their passions are and ask them about it.  Listening is the best way to engage.

Mike Albert - for example - keep checking with 'active' buyers on new listings on market and why it is good (or not so good) for them.

Face to face is best but if not possible you can still ask lots of questions and LISTEN.

I ask questions...and LISTEN!

Eye contact and a firm hand shake.

I like Kathleen Daniels response.....

Listen, listen, and look for opportunities to share solutions.

   Tell me more, Mike Albert.  What do mean when you say, "Engage"?  And what sort of examples do you need?

Once someone starts talking, become the best listener, coach & support U can