Nicky Syratt signed his two dogs over to the centre  in Aberdeenshire in June, but became upset when he discovered one had been put to sleep.
Police arrived to find him holding the other dog, and with a pair of scissors in his hand.
Syratt, 31, from Ellon, admitted threatening or abusive behaviour when he appeared at Aberdeen Sheriff Court.
The animal charity said the dog was put to sleep because it had an incurable disease.
Depute fiscal Katie Begg told the court that one of the rescue centre employees had been tasked with keeping Syratt informed about the progress of his dogs.
She said: "She did this by telephone and was in regular contact with him. One of the dogs had to be put down due to an incurable illness."
She said the employee contacted Syratt to let him know and he began to shout and swear at her, stating: "I'm coming down and nothing will stop me" and said he threatened to "do in staff".
Some of the staff were then sent home and the building was locked up as a result of the call.
One of the employees returned to her house opposite the rescue centre in Drumoak and saw Syratt arrive at the building later in the day.
Ms Begg said: "She then saw him tying a rope to the rear of his car to the door handle of the locus and he then accelerated forward in an attempt to remove the door."
After realising he had pulled the door handles off, Syratt then wandered into the building and forced his way into the cattery and kennels.
Two police officers who arrived at the scene shortly afterwards found him in a central courtyard with his dog, and holding scissors.
Ms Begg added: "One of the police officers asked the accused if the dog was dangerous, to which he replied, 'No but I am'."
He later dropped the scissors, as requested.
Defence lawyer Chris Maitland said his client denied threatening staff during the phone call.
He said Syratt suffered a head injury in a road accident 10 years ago which may have changed his personality.
He had a condition called Dissocial Personality Disorder for which there was no cure, he said.
The sheriff ordered Syratt to carry 80 hours of unpaid work.

People regularly playing sport stands at 15.74 million over 12 months, a rise of 1.65 million since 2005/06.
Bucking the normal trend, the number of women playing sport and being active is increasing faster than men.
Swimming remains the top sport overall but continues to decline, with athletics a close second and rising.
Sport England's Active People Survey began in 2005/06 and measures the number of people taking part in sport across the nation, with findings published every six months.
There was good news for tennis, cricket, boxing and rugby union, with increases in the numbers taking part but football and cycling experienced falls.
Participation figures for lower socio-economic groups and disabled people were low and remain static.
"It's good news that more women are playing sport, and driving an overall increase in the numbers," said Jennie Price, Sport England's chief executive.
"It's particularly great to see This Girl Can is making a real difference. That's because we've really focused on what drives women's attitudes and behaviours.
"If we're to see a further step change in the total number of people playing sport, we need to take a similar, consumer-focused approach in areas where the figures are stubbornly low, like disability and lower socio-economic groups."
A man forced his way into a Scottish SPCA rehoming centre after learning his dog had been put down, a court heard.
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An increase in the number of women playing sport has driven improved participation figures, according to the latest survey by Sport England.