hi,
a strawberry looks like a heart don't you think! so fits with the theme of your dream,
walking past a fence,this may refer to getting passed a barrier,have you been sitting on the fence about some issue or this someone,ie,not quite sure what to do,so avoiding making the move,
you arrive at the green grass,an area of growth,green to me is a colour of life, growth and possibility,
he gives you a green strawberry,why green? it is not ripe,you give him a red juicy delicious strawberry,red is again a colour of life vigour & passion,this suggests to me possibility again,all fruits ripen given the right conditions,and the dream may be showing you this potential,as it filled you with optimism on waking.
a great mentor of mine joseph campbell ,once said,
if you go where your heart & soul truly want to go,doors will open where you never expected to find any.
you sound like a lovely lady who deserves a break,i hope you find what you seek.
hope this helps.
rgds mikey